Grade 6 Workbook — Chapter 7: Exponents, Perfect Squares, and Powers of Ten

SOL 6.NS.3 · Companion to Textbook Chapter 7

Each page below is one Canva page. Headings are sized for direct paste: page title as H1, section labels as H2. Figures referenced by filename live in ../figures/.

PAGE 1 — Chapter opener

Chapter 7 · Exponents, Perfect Squares, and Powers of Ten

Standard 6.NS.3

In this chapter you will:

Write and evaluate powers with whole number bases and exponents
Find patterns in tables of powers
Model perfect squares with square arrays and justify whether a number is a perfect square
Connect powers of 10 to the place-value chart

Words to know: base · exponent · power · exponential form · expanded form · squared · cubed · perfect square · square array

The perfect squares in this chapter go up to $400$ . Learning them by sight makes every page faster.

PAGE 2 — Parts of a power

7.1 Exponents and Repeated Multiplication

FIGURE: fig1-exponent-parts.png (full width) [The expression 3 to the fourth power with arrows labeling the base and the exponent, and the expanded form 3 x 3 x 3 x 3 = 81 below it.]

The exponent tells how many FACTORS of the base, not what to multiply the base by.

Complete the table.

Exponential form	Expanded form	Value
$2^4$
$5^2$
$3^3$
$10^2$
$6^2$
$2^7$

True or false? $3^4 = 12$ ______ If false, write the correct value: ______

PAGE 3 — Evaluate and compare

Evaluating Powers

FIGURE: fig2-cube-model.png (right half of page) [A 5 by 5 grid of unit squares labeled 5 squared = 25, next to a cube of 8 unit cubes labeled 2 cubed = 8.]

Evaluate.

a) $2^3$	b) $3^2$	c) $4^2$	d) $5^3$
e) $7^2$	f) $10^4$	g) $1^6$	h) $9^1$

Fill in $<$ , $>$ , or $=$ .

a) $2^4 \ \square \ 4^2$	b) $3^3 \ \square \ 4^2$
c) $5^2 \ \square \ 2^5$	d) $6^2 \ \square \ 3^4$

Apply it. A bacterium doubles every hour. Starting from 1, the count after 6 hours is ______ , which equals ______ .

Explain. Why are $3^4$ and $4^3$ different?

PAGE 4 — Exit ticket 7.1

Exit Ticket · Lesson 7.1

Name: ________________________ Date: ____________

$4^3 =$ ______
Write $8 \times 8 \times 8$ using an exponent. ______
Which is greater, $5^2$ or $2^5$ ? ______
In $7^3$ , the base is ______ and the exponent is ______ . What does each one tell you?

PAGE 5 — Power tables

7.2 Patterns with Bases and Exponents

FIGURE: fig3-powers-of-two-pattern.png (full width) [A staircase of bars for 2 to the 1st through 2 to the 6th showing 2, 4, 8, 16, 32, 64, with arrows labeled "x 2" going right and "divide by 2" going left.]

Complete the tables.

$n$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$
$2^n$

$n$	$1$	$2$	$3$	$4$	$5$
$3^n$

$n$	$1$	$2$	$3$	$4$
$4^n$

$n$	$1$	$2$	$3$	$4$
$5^n$

The rule. To move one exponent up, ____________ by the base. To move one exponent down, ____________ by the base.

PAGE 6 — Ones-digit patterns

Predicting the Ones Digit

List the ones digit of each power of 2.

Power	$2^1$	$2^2$	$2^3$	$2^4$	$2^5$	$2^6$	$2^7$	$2^8$
Ones digit

The cycle repeats every ______ powers. Ones digit of $2^{12}$ : ______

Continue each pattern downward.

$4^3 = 64$ , $4^2 =$ ______ , $4^1 =$ ______ , $4^0 =$ ______

$10^3 = 1{,}000$ , $10^2 =$ ______ , $10^1 =$ ______ , $10^0 =$ ______

Explain. Why does every power of 5 end in the digit 5?

Apply it. A tournament of 64 teams cuts the field in half each round. $64 =$ ______ as a power of 2, so there are ______ rounds.

PAGE 7 — Exit ticket 7.2

Exit Ticket · Lesson 7.2

Name: ________________________ Date: ____________

$3^1 =$ ______ $3^2 =$ ______ $3^3 =$ ______ $3^4 =$ ______
Given $2^{10} = 1{,}024$ , find $2^{11}$ . ______
Ones digit of $9^3$ : ______
Describe the rule that moves from one power of a base to the next power up.

PAGE 8 — Square arrays

7.3 Perfect Squares

FIGURE: fig4-square-arrays.png (full width) [Square arrays of 1, 4, 9, 16, and 25 unit squares with the multiplication under each.]

A perfect square is a whole number of tiles that forms a square array with equal rows and columns.

Complete the list of perfect squares to 400.

$n$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$n^2$

$n$	$11$	$12$	$13$	$14$	$15$	$16$	$17$	$18$	$19$	$20$
$n^2$

FIGURE: fig5-odd-number-growth.png (right half of page) [A 5 by 5 grid with L-shaped bands shaded and labeled +3, +5, +7, +9 around a single shaded corner square.]

Fill in the gaps between consecutive perfect squares.

$1 \to 4$ : ______ $4 \to 9$ : ______ $9 \to 16$ : ______ $16 \to 25$ : ______ $25 \to 36$ : ______

The gaps are the consecutive ____________ numbers.

PAGE 9 — Justify it

Is It a Perfect Square?

To justify YES: name the whole number that squares to it. To justify NO: trap it between two consecutive perfect squares.

Complete the table.

Number	Perfect square?	Justification
$64$
$75$
$121$
$200$
$361$
$350$

List every perfect square between 100 and 200. _______________________________________________

Apply it. A square patio uses 169 tiles with none left over.

Tiles along one side: ______ Tiles around the outside border: ______

PAGE 10 — Exit ticket 7.3

Exit Ticket · Lesson 7.3

Name: ________________________ Date: ____________

$16^2 =$ ______
Is $196$ a perfect square? ______ Justification: _______________________
Is $300$ a perfect square? ______ Justification: _______________________
How does a square array show whether a number is a perfect square?

PAGE 11 — Powers of ten in the place-value chart

7.4 Powers of Ten and Place Value

FIGURE: fig6-place-value-powers.png (full width) [A place-value chart from millions to ones showing each place value as a number and as a power of ten, with arrows labeled "x 10" pointing left between columns.]

Complete the table.

Power	$10^0$	$10^1$	$10^2$	$10^3$	$10^4$	$10^5$	$10^6$
Value
Zeros

Write each as a power of ten.

$1{,}000 =$ ______ $10{,}000{,}000 =$ ______ $100 =$ ______ $1 =$ ______

Multiply.

$8 \times 10^3 =$ ______ $34 \times 10^2 =$ ______ $5 \times 10^5 =$ ______

PAGE 12 — Expanded form with powers of ten

Expanded Form

$3{,}507 = 3 \times 10^3 + 5 \times 10^2 + 0 \times 10^1 + 7 \times 10^0$

Write each number in expanded form using powers of ten.

$8{,}304 =$ _______________________________________________
$5{,}140 =$ _______________________________________________
$906 =$ _______________________________________________
$72{,}000 =$ _______________________________________________

Compare. Which is greater, $10^4$ or $4^{10}$ ? ______ Values: ______ and ______

Apply it. A stadium seats $5 \times 10^4$ people. Standard form: ______ Compared with $10^5$ , that is ____________ .

Explain. Why does the exponent on a power of 10 tell you the number of zeros?

PAGE 13 — Exit ticket 7.4

Exit Ticket · Lesson 7.4

Name: ________________________ Date: ____________

$10^3 =$ ______
Write $1{,}000{,}000$ as a power of ten. ______
$2{,}750$ in expanded form with powers of ten:

How does the place-value chart show that each place is 10 times the place to its right?

PAGE 14 — Chapter 7 review, part 1

Chapter 7 Review

Part A · Patterns with bases and exponents

Evaluate: a) $2^6$ ______ b) $3^4$ ______ c) $5^3$ ______
Write $7 \times 7 \times 7 \times 7$ as a power: ______ Value: ______
Given $2^7 = 128$ , find $2^8$ : ______
Least to greatest: $3^3$ , $2^5$ , $4^2$ , $5^2$

Part B · Perfect squares with models

List every perfect square from 1 to 100.
A square array uses 144 tiles. Tiles along one side: ______
Gap between $16$ and $25$ : ______ Why the model gives that gap:

PAGE 15 — Chapter 7 review, part 2

Chapter 7 Review (continued)

Part C · Justifying perfect squares

Is $289$ a perfect square? ______ Justification: _______________________
Is $130$ a perfect square? ______ Justification: _______________________
Perfect squares between 300 and 400: _______________________

Part D · Powers of ten and place value

$10^4 =$ ______ $10^0 =$ ______
$3{,}206$ in expanded form with powers of ten:
$62 \times 10^3 =$ ______

Part E · Application and reasoning

Which is greater, $6^3$ or $3^6$ ? ______ Values: ______ and ______
A square garden has area 225 sq ft. Side length: ______ Perimeter: ______
Is every power of 10 a perfect square? ______

Justify with examples: _______________________________________________

Canva production notes

Page size: 8.5 × 11 in, 0.75 in margins
Type: headings 24–28 pt, body 12–14 pt, answer blanks 14 pt with 1.5 line spacing
Exponents: set as true superscripts, at least 60 percent of body size, so they do not read as multiplication
Figure widths: fig1-exponent-parts.png, fig3-powers-of-two-pattern.png, fig4-square-arrays.png, and fig6-place-value-powers.png full width; fig2-cube-model.png and fig5-odd-number-growth.png half width
Grid work: page 8 needs a blank 20 × 20 grid in the margin so students can shade arrays while completing the perfect-square table
Answer blanks: keep every blank on the same line as its prompt so the Canva text box does not reflow