A square number is what you get when you multiply a whole number by itself, and a square root undoes that, telling you which number was squared. This page covers perfect squares, square roots, estimating roots that aren’t perfect squares, and practice.
What is a perfect square?
A perfect square is the result of multiplying a whole number by itself (Britannica). Squaring a number means raising it to the second power, written with a small 2, like 6².
For example, 6² means 6 x 6, which equals 36. So 36 is a perfect square. The first several perfect squares are 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196 and 225, from squaring 1 through 15.
Squares also connect to area. If a square has sides of 6 cm, its area is 6 x 6 = 36 square cm, the same multiplication used in our area formulas page.
This section covers squaring whole numbers. It doesn’t cover squaring fractions or negative numbers, which follow the same rule but need extra care with signs.
What is a square root?
A square root is a number that, when multiplied by itself, gives back the original number. It’s the reverse operation of squaring. The symbol for square root is the radical sign, √, and it dates back hundreds of years.
Since 6 x 6 = 36, the square root of 36 is 6, written √36 = 6. Every perfect square has a whole-number square root.
Some square roots.
- √36 = 6
- √49 = 7
- √64 = 8
- √81 = 9
- √100 = 10
This section covers finding the square root of a perfect square. It doesn’t cover square roots of negative numbers, which aren’t real numbers and belong in a later math class.
How do you estimate a square root that isn’t a perfect square?
Most numbers aren’t perfect squares, so their square roots aren’t whole numbers. You can still estimate one by finding the two perfect squares it falls between.
Worked example. Estimate √50.
- Find the closest perfect square below 50: 7² = 49.
- Find the closest perfect square above 50: 8² = 64.
- Since 49 < 50 < 64, the square root of 50 is between 7 and 8.
- Because 50 is much closer to 49 than to 64, √50 is closer to 7 than to 8. The actual value, rounded to two decimal places, is 7.07.
Another example. Estimate √90. The closest perfect squares are 9² = 81 and 10² = 100. Since 81 < 90 < 100, √90 is between 9 and 10, and closer to 10 since 90 is nearer 100. The actual value is about 9.49.
This section covers estimating square roots between two whole numbers by hand. It doesn’t cover finding exact decimal square roots, which normally uses a calculator.
Practice set
Work through these on paper, then check your answers.
- What is 9²?
- What is √81?
- What is 11²?
- Between which two whole numbers is √20?
- Between which two whole numbers is √110, and which one is it closer to?
- If a square has an area of 144 square cm, how long is each side?
Show answers and reasoning
- 9² = 9 x 9 = 81.
- √81 = 9, since 9 x 9 = 81.
- 11² = 11 x 11 = 121.
- √20 is between 4 and 5, since 4² = 16 and 5² = 25, and 16 < 20 < 25.
- √110 is between 10 and 11, since 10² = 100 and 11² = 121. It’s closer to 10, since 110 is nearer 100 than 121.
- The side length is √144 = 12 cm, since 12 x 12 = 144.
Squares and square roots come up constantly in geometry, especially for finding the side length of a square from its area, and they’re a building block for later topics like the Pythagorean theorem. They also connect to factoring, since finding a number’s square root is closely related to finding its factors, and to algebra, where exponents and roots show up again in equations. For more topics, visit our math hub.
Sources
- Britannica. Square root.