Square roots, radical notation, perfect squares intro
Middle School Bridge • 6-8
If you know that 52 = 25, then you also know that the square root of 25 is 5. Finding a square root is the reverse of squaring a number. This inverse relationship makes square roots essential in geometry (finding the side of a square from its area), physics, and countless other fields.
The square root of a number n is a value that, when multiplied by itself, gives n. The symbol for square root is the radical sign.
For example, √36 = 6 because 62 = 36.
A perfect square is a number whose square root is a whole number. Memorizing the first several perfect squares is extremely helpful.
| n | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| n2 | 1 | 4 | 9 | 16 | 25 | 36 | 49 | 64 | 81 | 100 | 121 | 144 |
Evaluate √81.
Ask: what number times itself equals 81? Since 9 × 9 = 81, √81 = 9.
Most numbers are not perfect squares. When you need √n and n is not a perfect square, you can estimate by finding which two consecutive perfect squares n falls between.
Estimate √50.
A radical expression can sometimes be simplified by factoring out a perfect square. Look for the largest perfect square factor of the number under the radical.
Simplify √72.
The square root of a sum is NOT the sum of the square roots. √(9 + 16) = √25 = 5, but √9 + √16 = 3 + 4 = 7. These are not equal! The radical sign applies to the entire expression under it, not to each piece separately.
To check if a radical is fully simplified, look at the number under the radical. If it has no perfect square factor other than 1, it is in simplest form. For example, √15 is already simplified because 15 = 3 × 5 and neither 3 nor 5 is a perfect square.
1. Evaluate √144.
12, because 122 = 144.
2. Between which two consecutive whole numbers does √30 fall?
Between 5 and 6, because 52 = 25 and 62 = 36, and 25 < 30 < 36.
3. Simplify √48.
48 = 16 × 3. √48 = √16 × √3 = 4√3.
4. Simplify √200.
200 = 100 × 2. √200 = √100 × √2 = 10√2.
5. A square has an area of 64 square meters. What is the side length?
Side = √64 = 8 meters.
The square root of a number n is the value that, when squared, gives n. Perfect squares have whole-number square roots. Non-perfect square roots can be estimated by locating them between consecutive perfect squares. Radicals can be simplified by factoring out perfect square factors. Remember that the square root of a sum is not the sum of the square roots.