Chapter 1 — Translating and Evaluating Algebraic Expressions

Standard: A.EO.1 — The student will represent verbal quantitative situations algebraically and evaluate these expressions for given replacement values of the variables.

Knowledge and Skills, quoted in full:

• a) Translate between verbal quantitative situations and algebraic expressions, including contextual situations.
• b) Evaluate algebraic expressions which include absolute value, square roots, and cube roots for given replacement values to include rational numbers, without rationalizing the denominator.

By the end of this chapter you will be able to:

• Translate a verbal phrase into an algebraic expression, and an algebraic expression back into words (A.EO.1a)
• Recognize the two phrases that reverse the order of the quantities they name (A.EO.1a)
• Define a variable for a contextual situation and write an expression that models it (A.EO.1a)
• Interpret what each number and each variable in a contextual expression means (A.EO.1a)
• Evaluate an algebraic expression for given replacement values, including rational numbers (A.EO.1b)
• Evaluate expressions containing absolute value, square roots, and cube roots (A.EO.1b)

Lessons: 1.1 The Language of Algebra · 1.2 Expressions from Contextual Situations · 1.3 Evaluating Expressions · 1.4 Absolute Value, Square Roots, and Cube Roots in Evaluation

Conventions this chapter fixes.

No rationalizing the denominator. A.EO.1b says so in as many words. If evaluating leaves a radical in a denominator, such as $\dfrac{6}{\sqrt{2}}$, that is the finished answer in this chapter. Rewriting it as $3\sqrt{2}$ is a legitimate technique, and Chapter 11 teaches it, but it is not asked for here and no answer in this chapter uses it.

Expressions, not equations. An expression is a phrase; an equation is a sentence. Nothing in this chapter has a solution, because nothing in this chapter is a question about an unknown. You will translate, interpret, and evaluate. Solving begins in Chapter 2.

The square root symbol asks for one number. $\sqrt{25}$ means $5$, never $\pm 5$. When a minus sign belongs to the answer, it is written outside the radical: $-\sqrt{25} = -5$.

Cube roots of negatives are real. $\sqrt[3]{-8} = -2$, because $(-2)^3 = -8$. There is no such thing as an even root of a negative real number, but odd roots of negatives are perfectly ordinary.

Calculator note. Algebra 1 has no no-calculator standards, and the Desmos Virginia calculator is available for the whole End-of-Course test. A calculator is still no help on the half of this chapter that matters most: it cannot decide whether "$7$ less than a number" means $n - 7$ or $7 - n$. Use it to check a substitution, not to make one. The perfect squares worth knowing on sight are $1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225$, and the perfect cubes are $1, 8, 27, 64, 125, 216, 343, 1000$.

Numbering note. Item numbers run straight through the chapter, from 1 in Lesson 1.1 to 122 at the end of the review. They do not restart at each lesson.

Lesson 1.1 — The Language of Algebra

The parts of an expression

An algebraic expression is a combination of numbers, variables, and operations that names a quantity. A variable is a letter standing for a number that is unknown or that is allowed to change. A numerical expression has no variable at all: $3 + 4 \cdot 5$ is numerical, and $3x + 4$ is algebraic.

An expression is built from terms, the pieces separated by $+$ and $-$ signs. In

$$5x^2 - 8x + 3$$

there are three terms: $5x^2$, $-8x$, and $3$. The number multiplying a variable in a term is the coefficient of that term, so the coefficient of $5x^2$ is $5$ and the coefficient of $-8x$ is $-8$. A term with no variable, here $3$, is a constant, because its value never changes no matter what $x$ is.

Notice that the sign travels with the term. The middle term is $-8x$, not $8x$. Reading a subtraction as "plus a negative" — $5x^2 + (-8x) + 3$ — makes that automatic and prevents a sign error later when you evaluate.

Two shorthands are worth stating out loud, because nothing in the notation announces them:

• Multiplication is written without a symbol. $6n$ means $6 \cdot n$, and $2(x + 5)$ means $2$ times the whole quantity $x + 5$.
• Division is written as a fraction. "A number divided by $4$" is $\dfrac{n}{4}$, not $n \div 4$, in almost every algebraic setting.

Turning words into symbols

A verbal quantitative situation is a description in words of a quantity or a relationship between quantities. Translating one is a matter of finding the operation words and the order they impose.

Most phrases are written in the order you hear them. "The sum of a number and $12$" is $n + 12$. "Three times a number" is $3n$. "A number divided by $7$" is $\dfrac{n}{7}$.

Two phrases are not, and they are the source of most translation errors in Algebra 1.

"Less than" reverses the order. "Five less than twice a number" is $2n - 5$, not $5 - 2n$.

The reason is what the words mean, not a rule to memorize. "Five less than twelve" is $7$: you start at twelve and go down five. Nobody reads that phrase as $5 - 12 = -7$. Algebra keeps the same meaning, so "five less than twice a number" starts at $2n$ and goes down five.

"Subtracted from" reverses the order too. "Nine subtracted from a number" is $n - 9$. The thing named right after from is what you start with.

Compare those with "the difference of $12$ and a number," which is written in the order you hear: $12 - n$. Subtraction is not commutative, so these are genuinely different expressions, and testing a phrase with an actual number is the fastest way to be sure which you have.

Where the parentheses come from

Parentheses appear when a phrase names an operation on a whole quantity rather than on a single number. The word to watch for is the sum, the difference, the quantity, or a comma.

• "Three times the sum of a number and $8$" — the tripling applies to the whole sum: $3(n + 8)$.
• "Three times a number, plus $8$" — the tripling applies only to the number: $3n + 8$.
• "The square of the sum of a number and $3$" — $(n + 3)^2$.
• "The sum of the square of a number and $3$" — $n^2 + 3$.

Those four expressions are all different, and the only thing separating them is which quantity the operation is applied to.

Reading an expression back into words

Translation runs both directions, and A.EO.1a asks for both. Given $4n - 1$, a correct verbal phrase is "one less than four times a number," or "the product of $4$ and a number, decreased by $1$." More than one phrasing is right, as long as the operations and their order come out the same.

Worked examples

Example 1 — A phrase written in order

Translate "the sum of a number and $15$."

Sum signals addition, and the phrase names the number first.

Answer: $n + 15$

Example 2 — A phrase that reverses

Translate "$9$ less than a number."

Less than means start with the second quantity named and subtract the first. Start at $n$, take away $9$. Check with a real number: $9$ less than $20$ is $11$, and $20 - 9 = 11$, so the order is right.

Answer: $n - 9$

Example 3 — Multiplication and a following operation

Translate "twice a number, decreased by $7$."

Twice is multiplication by $2$, giving $2n$. Decreased by is subtraction applied after.

Answer: $2n - 7$

Example 4 — A phrase that needs parentheses

Translate "three times the sum of a number and $8$."

The quantity being tripled is the entire sum $n + 8$, so it must be grouped. Without the parentheses, $3n + 8$ would triple only the number.

Answer: $3(n + 8)$

Example 5 — Naming the parts

In $7x^2 - 3x + 10$, name the number of terms, the coefficients, and the constant.

Separate at the $+$ and $-$ signs, keeping each sign with the term that follows it: $7x^2$, $-3x$, and $10$.

Answer: three terms; coefficients $7$ and $-3$; constant $10$.

Example 6 — Going from symbols to words

Write a verbal phrase for $\dfrac{n + 5}{2}$.

The fraction bar groups, so the sum happens first and the division second.

Answer: "the sum of a number and $5$, divided by $2$" — or "half of the sum of a number and $5$."

Guided practice

Let $n$ represent the number in items 1 through 6.

1. Translate: the sum of a number and $15$.
2. Translate: $9$ less than a number.
3. Translate: the product of $6$ and a number.
4. Translate: the quotient of a number and $4$.
5. Translate: twice a number, decreased by $7$.
6. Translate: $5$ more than three times a number.
7. Write a verbal phrase for $n + 10$.
8. Write a verbal phrase for $4n - 1$.

Independent practice

9. Translate each phrase. Only two different expressions appear among the five; say which phrases produce each one. a) the difference of a number and $12$ b) $12$ less than a number c) the difference of $12$ and a number d) a number subtracted from $12$ e) $12$ subtracted from a number
10. Translate each phrase. a) one-third of a number b) the quotient of $20$ and a number c) a number divided by $20$ d) the sum of a number and its square e) the square of the sum of a number and $3$
11. Translate each phrase, using parentheses only where the phrase requires them. a) three times the sum of a number and $8$ b) three times a number, plus $8$ c) the sum of $8$ and three times a number d) $8$ less than the product of $3$ and a number
12. Write a verbal phrase for each expression. a) $7n$ b) $n - 11$ c) $\dfrac{n + 5}{2}$ d) $2n + 9$ e) $6(n - 4)$
13. For the expression $5x^2 - 8x + 3$: a) how many terms are there? b) what is the coefficient of the first term? c) what is the coefficient of the second term? d) what is the constant?
14. Copy and complete the table.

15. Which of these expressions matches "ten decreased by the product of $4$ and a number"? $4n - 10$, $10 - 4n$, $4(10 - n)$, $\dfrac{10}{4n}$. Explain how you eliminated the other three.
16. Application. A parking garage holds $n$ cars at the start of the hour, and $12$ cars leave during that hour. Write an expression for the number of cars remaining, and explain why the expression is not $12 - n$.
17. Reasoning. Explain why "$7$ less than a number" is $n - 7$ rather than $7 - n$. Support your explanation by replacing "a number" with $20$ and evaluating both.
18. Error analysis. A student translates "$5$ less than the product of $3$ and a number" as $5 - 3n$. Identify the error, give the correct expression, and describe a check the student could have done in one line.

Exit ticket 1.1

19. Translate: $10$ less than twice a number.
20. Translate: the quotient of a number and $6$, increased by $5$.
21. Write a verbal phrase for $3(n - 2)$.
22. Explain, in words, the difference between $4n - 9$ and $9 - 4n$.

Lesson 1.2 — Expressions from Contextual Situations

Deciding what the variable stands for

A contextual situation is a real setting — a price, a distance, a temperature, a count — described in words. Modeling one algebraically is a three-step habit, and the first step is the one students skip.

• First, define the variable. Write a sentence: "Let $h$ = the number of hours worked." A letter with no sentence attached is not yet a variable; it is a mystery.
• Next, find what changes and what does not. A quantity that is the same no matter what happens is a constant. A quantity that depends on the variable is multiplied by it.
• Last, write the expression and read it back against the situation to check it.

Say what the variable measures, not what object it is. "Let $h$ = hours" is loose; "let $h$ = the number of hours worked" is a number, and only numbers can be multiplied by $15$.

The pattern behind most contextual expressions

An enormous fraction of real situations have the same shape: something charged once, plus something charged per unit.

The $\$25$ fee appears once in the picture no matter how many hours are worked, so it appears once in the expression as a constant. The $\$15$ appears once per hour, so it is multiplied by the number of hours. The result is

$$25 + 15h, \qquad \text{where } h = \text{the number of hours worked.}$$

Read the expression back the other way and you have interpreted it, which A.EO.1a also asks for: in $25 + 15h$, the $25$ is the one-time fee in dollars, the $15$ is the hourly rate in dollars per hour, and $h$ is the number of hours.

Two variations on the same shape come up constantly.

• A starting amount that decreases. A tank holds $80$ gallons and drains $6$ gallons per minute: $80 - 6m$. The rate is subtracted because the quantity is going down.
• Two different rates. $q$ quarters and $d$ dimes are worth $25q + 10d$ cents. Each count gets its own rate, and the two products are added.

Multiple quantities described in terms of one

When a situation describes two related quantities, choose the variable for the one everything else is described in terms of, then build the others from it.

A rectangle's length is $4$ more than its width. Let $w$ = the width in centimeters. Then the length is $w + 4$, and the perimeter is

$$2w + 2(w + 4) = 2w + 2w + 8 = 4w + 8.$$

Had you started with the length, you would have had to describe the width as "$4$ less than the length," which is correct but does more work.

Worked examples

Example 1 — A single rate

A movie ticket costs $\$12$. Write an expression for the cost of $t$ tickets.

Let $t$ = the number of tickets. Each ticket adds $\$12$, so the cost is $12$ multiplied by the count.

Answer: $12t$ dollars

Example 2 — A constant plus a rate

A pizza costs $\$14$ plus $\$2$ for each topping. Write an expression for the cost with $p$ toppings.

Let $p$ = the number of toppings. The $\$14$ is charged once; the $\$2$ is charged once per topping.

Answer: $14 + 2p$ dollars

Example 3 — A starting amount that decreases

The temperature is $68\,^\circ\text{F}$ and falls $3\,^\circ\text{F}$ each hour. Write an expression for the temperature after $h$ hours.

Let $h$ = the number of hours. Falling means the accumulated change is subtracted from the start.

Answer: $68 - 3h$ degrees Fahrenheit

Example 4 — Sharing equally

A class of $s$ students is divided into $5$ equal groups. Write an expression for the number of students in each group.

Divided into equal groups is division, and the total is what gets divided.

Answer: $\dfrac{s}{5}$ students

Example 5 — Building a second quantity from the first

A rectangle's length is $4$ centimeters more than its width. Write expressions for the length and for the perimeter.

Let $w$ = the width in centimeters. The length is $w + 4$. Perimeter is twice the width plus twice the length:

$$2w + 2(w + 4) = 2w + 2w + 8 = 4w + 8$$

Answer: length $w + 4$ cm; perimeter $4w + 8$ cm

Example 6 — Interpreting an expression someone else wrote

A plumber's charge in dollars is $45 + 12n$. What does each part of the expression mean?

The $12$ is attached to $n$, so it is a rate; the $45$ stands alone, so it is charged once.

Answer: $\$45$ is a one-time service charge, $\$12$ is the charge per unit of $n$ (per quarter-hour, per hour, or whatever $n$ counts), and $n$ is the number of those units.

Guided practice

For items 23 through 29, define the variable in a sentence and then write the expression.

23. A movie ticket costs $\$12$. Write an expression for the cost of $t$ tickets.
24. A pizza costs $\$14$ plus $\$2$ for each topping. Write an expression for the cost with $p$ toppings.
25. Sara has $\$50$ and spends some of it. Write an expression for the amount she has left.
26. A class of $s$ students is divided into $5$ equal groups. Write an expression for the size of each group.
27. A rectangle's length is $4$ centimeters more than its width $w$. Write an expression for the length.
28. Write an expression for the perimeter of the rectangle in item 27, simplified.
29. A gym charges a $\$30$ joining fee plus $\$22$ per month. Write an expression for the total cost after $m$ months.

Independent practice

30. Notebooks cost $\$3.25$ each. a) Write an expression for the cost of $n$ notebooks. b) Write an expression for the cost of $n$ notebooks plus one $\$5$ bag.
31. A car travels at $55$ miles per hour. a) Write an expression for the distance covered in $h$ hours. b) Write an expression for the total distance if the driver had already gone $20$ miles before starting the timer.
32. A jar holds $q$ quarters and $d$ dimes. a) Write an expression for the total value in cents. b) Write an expression for the total value in dollars.
33. A tank holds $80$ gallons and drains at $6$ gallons per minute. Write an expression for the number of gallons left after $m$ minutes.
34. In the expression $40 + 0.25g$ for a phone bill in dollars, where $g$ is the number of gigabytes used beyond the plan limit, explain what the $40$, the $0.25$, and the $g$ each represent.
35. Match each situation to its expression: $8n$, $8 + n$, $n - 8$, $\dfrac{n}{8}$, $8 - n$. a) $n$ boxes of $8$ pencils b) $n$ dollars after spending $\$8$ c) $n$ cookies shared equally by $8$ people d) $8$ dollars plus $n$ dollars e) $8$ feet of ribbon with $n$ feet cut off
36. Write an expression for each, defining the variable first. a) the number of days in $w$ weeks b) the number of inches in $f$ feet c) the age in years of someone $5$ years younger than a person who is $a$ years old
37. Application. A theater has some number of rows with $18$ seats in each row, plus $12$ balcony seats. Define a variable and write an expression for the total number of seats. Then say what your expression would become if the balcony were closed.
38. Reasoning. Explain why $5 + 3n$ and $3n + 5$ describe the same situation, but $5 - 3n$ and $3n - 5$ do not. Use the properties of the operations in your explanation.
39. Error analysis. A jacket costs $c$ dollars, and a shirt costs $\$20$ less than the jacket. A student writes the shirt's cost as $20 - c$. Explain the error, give the correct expression, and explain what the student's expression would describe instead.

Exit ticket 1.2

40. A taxi charges $\$3.50$ plus $\$2$ per mile. Define a variable and write an expression for the fare.
41. A bag of $s$ marbles is shared equally among $6$ friends. Write an expression for each friend's share.
42. Describe a real situation that the expression $8n + 15$ could model. Say what $n$ counts and what the $8$ and the $15$ mean.
43. Explain how you decide which quantity in a situation should be represented by the variable.

Lesson 1.3 — Evaluating Expressions

Substitution, then arithmetic

To evaluate an algebraic expression is to replace each variable with a given replacement value and simplify the resulting numerical expression to a single number. There is exactly one thing to get right in the substitution step and one thing to get right afterward.

Substitute inside parentheses, always. Replacing $x$ with $-2$ in $3x^2 - 4x$ gives $3(-2)^2 - 4(-2)$. Writing $3-2^2$ instead loses both the multiplication and the sign, and there is no recovering from it.

Then follow the order of operations. Grouping symbols first, then exponents and roots, then multiplication and division left to right, then addition and subtraction left to right.

Two sign situations account for most wrong answers on this skill.

$-x^2$ and $(-x)^2$ are different expressions. The exponent attaches only to what it sits on. In $-6^2$ the exponent sits on the $6$, so you square first and negate second: $-36$. In $(-6)^2$ the exponent sits on the whole parenthesized quantity: $36$.

Subtracting a negative. In $12 - (-8)$, the parentheses from the substitution are what let you see that this is a subtraction of a negative, which adds: $12 + 8 = 20$.

Rational replacement values

A.EO.1b requires replacement values that include rational numbers, which means fractions and decimals, positive and negative. Nothing about the procedure changes; only the arithmetic gets more careful.

Evaluate $2p - 3q$ for $p = \tfrac{1}{2}$ and $q = -\tfrac{2}{3}$:

$$2\left(\frac{1}{2}\right) - 3\left(-\frac{2}{3}\right) = 1 - (-2) = 1 + 2 = 3$$

A fraction replacement value is often a gift rather than a burden. A coefficient of $6$ meeting a replacement value of $\tfrac{2}{3}$ produces the integer $4$, and problems are frequently built that way on purpose.

Absolute value

The absolute value of a number is its distance from $0$ on the number line, written with bars: $|{-6}| = 6$ and $|6| = 6$.

Because a distance is never negative, an absolute value is never negative. But "never negative" is a fact about what is inside the bars, and two consequences follow that students routinely miss.

The bars are grouping symbols. Everything inside gets simplified before the absolute value is taken. In $|2x + y|$ with $x = -5$ and $y = 3$, first get $|{-7}|$, then $7$.

A minus sign outside the bars survives. $-|{-6}| = -6$. The bars turned $-6$ into $6$; the minus sign in front then turns it back. Only what is inside the bars is affected.

One more distinction worth practicing, because it appears on tests: $|7 - 15| = |{-8}| = 8$, but $|7| - |15| = 7 - 15 = -8$. Where the bars fall changes the answer.

Worked examples

Example 1 — An integer replacement value

Evaluate $4x + 9$ for $x = -3$.

$$4(-3) + 9 = -12 + 9 = -3$$

Answer: $-3$

Example 2 — An exponent and a negative value

Evaluate $3x^2 - 4x$ for $x = -2$.

$$3(-2)^2 - 4(-2) = 3(4) - 4(-2) = 12 - (-8) = 12 + 8 = 20$$

Answer: $20$

Example 3 — A fraction replacement value inside parentheses

Evaluate $5 - 2(m + 7)$ for $m = -\tfrac{1}{2}$.

$$5 - 2\left(-\frac{1}{2} + 7\right) = 5 - 2\left(\frac{13}{2}\right) = 5 - 13 = -8$$

Answer: $-8$

Example 4 — Absolute value with and without an outside sign

Evaluate $|x - 8|$ and $-|x - 8|$ for $x = 3$.

$$|3 - 8| = |{-5}| = 5 \qquad -|3 - 8| = -|{-5}| = -5$$

Answer: $5$ and $-5$

Example 5 — Two rational replacement values

Evaluate $2p - 3q$ for $p = \tfrac{1}{2}$ and $q = -\tfrac{2}{3}$.

$$2\left(\frac{1}{2}\right) - 3\left(-\frac{2}{3}\right) = 1 - (-2) = 3$$

Answer: $3$

Example 6 — Absolute value bars used twice

Evaluate $|2x + y| - |y|$ for $x = -5$ and $y = 3$.

$$|2(-5) + 3| - |3| = |{-10} + 3| - 3 = |{-7}| - 3 = 7 - 3 = 4$$

Answer: $4$

Guided practice

44. Evaluate $3x + 5$ for $x = 4$.
45. Evaluate $3x + 5$ for $x = -4$.
46. Evaluate $7 - 2n$ for $n = -3$.
47. Evaluate $x^2$ for $x = -6$.
48. Evaluate $-x^2$ for $x = -6$.
49. Evaluate $|a|$ for $a = -9$.
50. Evaluate $-|a|$ for $a = -9$.
51. Evaluate $|a - 4|$ for $a = -9$.
52. Evaluate $4c - 1$ for $c = \tfrac{1}{2}$.

Independent practice

53. Evaluate each for $x = -2$ and $y = 5$. a) $x + y$ b) $x - y$ c) $xy$ d) $y - x$ e) $3x + 2y$
54. Evaluate each for $a = -3$ and $b = 4$. a) $a^2 + b^2$ b) $(a + b)^2$ c) $2a^2 - b$ d) $-2ab$
55. Evaluate each for $m = \tfrac{2}{3}$. a) $6m$ b) $9m - 1$ c) $m + \tfrac{1}{6}$ d) $\dfrac{1}{m}$
56. Evaluate each for $t = -1.5$. a) $4t$ b) $t^2$ c) $10 - 8t$ d) $|t|$
57. Evaluate each. a) $|{-12}|$ b) $-|{-12}|$ c) $|7 - 15|$ d) $|7| - |15|$ e) $|{-3}| \cdot |{-4}|$
58. Evaluate $|2x - 7|$ for each replacement value. a) $x = 5$ b) $x = 1$ c) $x = 3.5$ d) $x = -2$
59. Evaluate $\dfrac{x + y}{x - y}$ for a) $x = 5$, $y = 3$ b) $x = \tfrac{1}{2}$, $y = \tfrac{1}{4}$
60. Evaluate $5 - 2(x - 3)^2$ for $x = 1$.
61. Application. The change in temperature between two readings $T_1$ and $T_2$ is the number of degrees between them, which is $|T_2 - T_1|$. A morning reading is $-4\,^\circ\text{F}$ and the afternoon reading is $11\,^\circ\text{F}$. Evaluate the expression, and explain why absolute value is the right tool for "how many degrees apart."
62. Application. A repair shop's charge in dollars is $25 + 15h$, where $h$ is the number of hours of work. Evaluate the expression for $h = 3.5$ and write the result as an amount of money.
63. Reasoning. Evaluate $(-6)^2$ and $-6^2$. Explain why the two results differ even though both expressions contain a $6$, a minus sign, and a square.
64. Error analysis. A student evaluates $5 - 2x$ for $x = -3$ and writes $5 - 6 = -1$. Find the step where the error happened, give the correct value, and state the rule the student missed.

Exit ticket 1.3

65. Evaluate each for $x = -5$. a) $2x - 9$ b) $x^2 + x$
66. Evaluate $6n + 1$ for $n = \tfrac{1}{3}$.
67. Evaluate $|x - 10|$ and $-|x - 10|$ for $x = 2.5$.
68. Explain why substituting a negative replacement value in parentheses matters. Use an example.

Lesson 1.4 — Absolute Value, Square Roots, and Cube Roots in Evaluation

Two more operations inside the same procedure

Nothing about evaluation changes in this lesson. You still substitute in parentheses and still follow the order of operations. What is new is that A.EO.1b names two operations that can appear in the expression you are evaluating.

The square root of a nonnegative number $a$, written $\sqrt{a}$, is the nonnegative number whose square is $a$. So $\sqrt{49} = 7$, because $7^2 = 49$.

The cube root of a number $a$, written $\sqrt[3]{a}$, is the number whose cube is $a$. So $\sqrt[3]{8} = 2$, because $2^3 = 8$.

The difference between them shows up as soon as the number underneath goes negative.

Cube roots of negative numbers are real numbers. $\sqrt[3]{-27} = -3$, because $(-3)^3 = -27$. Cubing preserves sign, so every real number — positive, negative, or zero — has exactly one real cube root.

Square roots of negative numbers are not real numbers. No real number squares to $-25$, because squaring a positive gives a positive and squaring a negative also gives a positive. If a substitution ever lands you at $\sqrt{-25}$, the expression has no real value for that replacement value, and saying so is the correct answer.

A minus sign in front of a radical is applied last. $-\sqrt{16} = -4$, because $\sqrt{16} = 4$ and then the minus sign is applied. That is a different expression from $\sqrt{-16}$, which is not real at all.

A radical is a grouping symbol. Everything under the bar is simplified before the root is taken. Evaluating $\sqrt{x + y}$ at $x = 40$ and $y = 9$ gives $\sqrt{49} = 7$, not $\sqrt{40} + \sqrt{9}$. Roots do not distribute over addition, and $\sqrt{9 + 16} = 5$ while $\sqrt{9} + \sqrt{16} = 7$ is the cleanest proof of it.

Rational numbers under the radical

Perfect squares and perfect cubes do not have to be whole numbers. Because $\left(\tfrac{3}{4}\right)^2 = \tfrac{9}{16}$, we get $\sqrt{\tfrac{9}{16}} = \tfrac{3}{4}$, and the same idea handles decimals: $\sqrt{2.25} = 1.5$ because $1.5^2 = 2.25$. For fractions, take the root of the numerator and the root of the denominator separately.

The convention this chapter fixes: no rationalizing

Sometimes a substitution leaves a radical in a denominator. Evaluate $\dfrac{6}{\sqrt{x}}$ at $x = 2$ and you get

$$\frac{6}{\sqrt{2}}$$

and in this chapter that is the finished answer. A.EO.1b says "without rationalizing the denominator," so you are not asked to rewrite it. The rewriting is a real technique — it produces $3\sqrt{2}$, the same number — and Chapter 11 develops it along with the rest of simplest radical form. Here, stop when the arithmetic is done.

If the replacement value happens to make the radical come out even, of course you finish the division: $\dfrac{6}{\sqrt{4}} = \dfrac{6}{2} = 3$.

Worked examples

Example 1 — A square root of a perfect square

Evaluate $\sqrt{x}$ for $x = 81$.

$81 = 9^2$, and the square root symbol asks for the nonnegative root.

Answer: $9$

Example 2 — The radical as a grouping symbol

Evaluate $\sqrt{x + y}$ for $x = 40$ and $y = 9$.

Simplify under the bar first:

$$\sqrt{40 + 9} = \sqrt{49} = 7$$

Evaluating the roots separately would give $\sqrt{40} + \sqrt{9} \approx 6.32 + 3 = 9.32$, a different number.

Answer: $7$

Example 3 — A cube root of a negative number

Evaluate $\sqrt[3]{n}$ for $n = -64$.

Ask what number cubed gives $-64$. Since $4^3 = 64$ and cubing preserves sign, $(-4)^3 = -64$.

Answer: $-4$

Example 4 — A root inside a larger expression

Evaluate $2\sqrt[3]{x} + 5$ for $x = 27$.

Roots are evaluated at the same stage as exponents, before the multiplication:

$$2\sqrt[3]{27} + 5 = 2(3) + 5 = 6 + 5 = 11$$

Answer: $11$

Example 5 — Three replacement values under one radical

Evaluate $\sqrt{b^2 - 4ac}$ for $a = 1$, $b = 7$, and $c = 6$.

$$\sqrt{(7)^2 - 4(1)(6)} = \sqrt{49 - 24} = \sqrt{25} = 5$$

Answer: $5$

Example 6 — A radical left in the denominator

Evaluate $\dfrac{6}{\sqrt{x}}$ for $x = 2$.

$$\frac{6}{\sqrt{2}}$$

Since $2$ is not a perfect square, $\sqrt{2}$ does not simplify to a rational number, and A.EO.1b does not ask you to rationalize the denominator.

Answer: $\dfrac{6}{\sqrt{2}}$

Guided practice

69. Evaluate $\sqrt{49}$.
70. Evaluate $\sqrt{\tfrac{1}{4}}$.
71. Evaluate $\sqrt[3]{27}$.
72. Evaluate $\sqrt[3]{-8}$.
73. Evaluate $\sqrt{x}$ for $x = 121$.
74. Evaluate $\sqrt[3]{x}$ for $x = -125$.
75. Evaluate $3\sqrt{x} - 2$ for $x = 16$.
76. Evaluate $\sqrt[3]{x} + |x|$ for $x = -27$.
77. Evaluate $\dfrac{10}{\sqrt{x}}$ for $x = 5$. Leave the denominator as it is.

Independent practice

78. Evaluate each. a) $\sqrt{64}$ b) $\sqrt{144}$ c) $\sqrt{\tfrac{9}{16}}$ d) $\sqrt{0}$ e) $\sqrt{1.44}$
79. Evaluate each. a) $\sqrt[3]{1}$ b) $\sqrt[3]{64}$ c) $\sqrt[3]{-27}$ d) $\sqrt[3]{-1}$ e) $\sqrt[3]{\tfrac{8}{27}}$
80. Evaluate $\sqrt{a + b}$ for each pair. a) $a = 7$, $b = 9$ b) $a = -11$, $b = 47$ c) $a = \tfrac{1}{2}$, $b = \tfrac{1}{2}$
81. Evaluate $\sqrt[3]{x - 1}$ for each replacement value. a) $x = 9$ b) $x = -7$ c) $x = 1$
82. Evaluate $\sqrt{b^2 - 4ac}$ for a) $a = 2$, $b = 7$, $c = 3$ b) $a = 1$, $b = 6$, $c = 5$
83. Evaluate $|x| + \sqrt{x}$ for a) $x = 9$ b) $x = \tfrac{1}{4}$
84. Evaluate $\dfrac{12}{\sqrt{x}}$ for a) $x = 3$ b) $x = 4$. One of these finishes as a rational number and one does not; say which is which and why.
85. Evaluate $2\sqrt{x} + \sqrt[3]{y}$ for $x = 25$ and $y = -8$.
86. Application. A square patio covers $A$ square feet, so each side measures $\sqrt{A}$ feet. A cube-shaped storage bin holds $V$ cubic inches, so each edge measures $\sqrt[3]{V}$ inches. Find the side of a patio with $A = 169$ and the edge of a bin with $V = 216$.
87. Application. An object dropped from a height of $d$ feet falls for $\sqrt{\dfrac{d}{16}}$ seconds before it lands. Evaluate the expression for $d = 144$ feet and state the result with its unit.
88. Reasoning. Explain why $\sqrt[3]{-64}$ is a real number but $\sqrt{-64}$ is not. Use the effect of squaring and of cubing on the sign of a number.
89. Error analysis. A student claims that $\sqrt{x^2 + y^2} = x + y$ and uses it to evaluate the expression at $x = 3$, $y = 4$, getting $7$. Evaluate the expression correctly, and explain what the student's shortcut assumes that is false.

Exit ticket 1.4

90. Evaluate a) $\sqrt{x}$ for $x = 100$ b) $\sqrt[3]{x}$ for $x = -216$
91. Evaluate $5\sqrt[3]{x} - \sqrt{y}$ for $x = 8$ and $y = 49$.
92. Evaluate $\dfrac{9}{\sqrt{x}}$ for $x = 6$, and explain why you are finished.
93. Explain why $\sqrt{-25}$ has no real value while $\sqrt[3]{-25}$ does.

Chapter 1 Review

Vocabulary. algebraic expression · numerical expression · variable · term · coefficient · constant · verbal quantitative situation · contextual situation · translate · evaluate · replacement value · absolute value · square root · cube root

Part A — Translating between verbal situations and algebraic expressions (A.EO.1a)

94. Translate each phrase, using $n$ for the number. a) the sum of a number and $12$ b) $8$ less than a number c) the quotient of a number and $5$ d) three times the difference of a number and $7$ e) twice a number, increased by $9$
95. Write a verbal phrase for each expression. a) $5n - 2$ b) $\dfrac{n + 4}{3}$ c) $4(n - 6)$
96. Explain the difference between "the difference of $6$ and a number" and "$6$ less than a number." Write both expressions.
97. Application. A cell plan costs $\$30$ per month plus $\$0.10$ for each text message. Define a variable and write an expression for the monthly bill in dollars.
98. Application. A rectangle's length is $3$ more than twice its width. Define a variable and write expressions for a) the length and b) the perimeter, simplified.
99. Application. Marcus has $d$ dollars and spends $\$18.75$. Write an expression for what he has left.
100. Application. A cash drawer holds $n$ nickels and $d$ dimes. Write an expression for the total value in cents.
101. Application. An auditorium has $r$ rows of $24$ seats, plus $8$ seats on the floor. Define the variable and write an expression for the total number of seats.
102. In the expression $7x^2 - 3x + 10$, name a) the number of terms b) the coefficient of the second term c) the constant.
103. Copy and complete the table.

104. Reasoning. Explain why "$5$ less than $x$" is $x - 5$ and not $5 - x$. Test both with $x = 12$ and report what each gives.
105. Error analysis. A student writes "the quotient of $8$ and a number" as $\dfrac{n}{8}$. Explain the error and give the correct expression.

Part B — Evaluating expressions, including absolute value and roots (A.EO.1b)

106. Evaluate each for $x = -4$ and $y = 2$. a) $3x + y$ b) $x^2 - y$ c) $xy^2$ d) $\dfrac{x + y}{2}$ e) $2(x - 3y)$
107. Evaluate each for $a = \tfrac{1}{2}$ and $b = -\tfrac{3}{4}$. a) $4a$ b) $8b$ c) $a + b$ d) $a - b$ e) $ab$
108. Evaluate each for $p = -2.5$. a) $4p$ b) $p^2$ c) $|p|$ d) $-|p|$ e) $6 - 2p$
109. Evaluate each. a) $|{-15}|$ b) $-|15|$ c) $|{-15} + 4|$ d) $|{-15}| + |4|$ e) $-|{-15} + 4|$
110. Evaluate $|3x - 5|$ for a) $x = 4$ b) $x = 0$ c) $x = \tfrac{5}{3}$ d) $x = -1$
111. Evaluate each. a) $\sqrt{196}$ b) $\sqrt{\tfrac{25}{49}}$ c) $\sqrt{2.25}$ d) $\sqrt{0}$
112. Evaluate each. a) $\sqrt[3]{125}$ b) $\sqrt[3]{-125}$ c) $\sqrt[3]{\tfrac{1}{8}}$ d) $\sqrt[3]{-1000}$
113. Evaluate $\sqrt{x + y}$ for a) $x = 52$, $y = 12$ b) $x = -9$, $y = 13$ c) $x = 0.75$, $y = 0.25$
114. Evaluate $2\sqrt[3]{x} + \sqrt{y}$ for $x = -27$ and $y = 36$.
115. Evaluate $\dfrac{14}{\sqrt{x}}$ for a) $x = 7$ b) $x = 49$. Do not rationalize any denominator.
116. Evaluate $\sqrt{b^2 - 4ac}$ for $a = 3$, $b = 11$, $c = 6$.
117. Application. A repair shop's charge in dollars is $25 + 15h$, where $h$ is the number of hours worked. Evaluate for $h = 2.5$ and write the result as an amount of money.
118. Application. A square rug covers $6.25$ square meters, so each side is $\sqrt{6.25}$ meters. Evaluate.
119. Application. A cubical shipping box holds $343$ cubic centimeters, so each edge is $\sqrt[3]{343}$ centimeters. Evaluate.
120. Application. Overnight the temperature was $-13\,^\circ\text{F}$ and by noon it was $9\,^\circ\text{F}$. The number of degrees between the two readings is $|T_2 - T_1|$. Evaluate it, and explain why the answer would be the same if the two readings were named in the other order.
121. Reasoning. Evaluate $x^2$ and $-x^2$ for $x = -5$. Explain why the two results differ, referring to what the exponent is attached to in each expression.
122. Error analysis. A student evaluates $\dfrac{20}{\sqrt{x}}$ for $x = 5$, writes $\dfrac{20}{\sqrt{5}}$, and then says the problem is not finished because the denominator still has a radical. Explain what this chapter's convention says about that, and name the chapter where the technique the student is thinking of is taught.

Standards coverage check — Chapter 1

Coverage notes. Rational replacement values appear in items 52, 55, 56, 58c, 59b, 66, 67, 70, 78c, 78e, 79e, 80c, 83b, 107, 108, 110c, 111b, 111c, 112c, and 113c. Absolute value appears in items 49–51, 56d, 57, 58, 61, 67, 76, 83, 108c, 108d, 109, 110, and 120. Square roots appear in items 69, 70, 73, 75, 77, 78, 80, 82–87, 89, 90a, 91, 92, 111, 113, 114, 115, 116, and 118. Cube roots appear in items 71, 72, 74, 76, 79, 81, 85, 86, 88, 90b, 91, 93, 112, 114, and 119. Denominators containing radicals are left unrationalized in items 77, 84a, 92, 115a, and 122.

This chapter stays inside A.EO.1. No expression here is set equal to anything, because solving equations is A.EI.1 and belongs to Chapter 2. Simplifying radicals into simplest radical form, including rationalizing denominators, is A.EO.4 and belongs to Chapter 11. Combining like terms and operating on polynomials is A.EO.2 and belongs to Chapter 12.

Answer keys for every set in this chapter are in Appendix A.