# Square Roots and Perfect Squares

The **square root** of a number is the reverse of squaring a number, or raising a number to the second power. The square root is the number that when multiplied by itself equals that value. For instance, the square root of 4 is 2 because 2 times itself is 4. 4 is a perfect square.

## What is a Perfect Square?

A **perfect square** is a value that has a whole number square root. The square root of 4 is 2, so its square root is a whole number, which means the four is a perfect square. Another example of a perfect square is 9.

The square root of 9 is 3. The square root is a whole number, which means that 9 is a perfect square. The square root is 3. This is not a perfect square, because the square root of 3 is not a whole number. The square root 3 is actually an irrational number.

You can find the value on your calculator, but it does go on forever and ever. Roots can be expressed in exponential form as fractions. The square root of 3 we could write is 3 to the 1/2 power. These are equivalent.

You could write, the cube root of 8 also as a fraction, as an exponent with a fraction, so that be 8 to 1/3 power. The cube root of eight, It’s very similar to finding the square root of a number, but when you find the cube root you’re finding what number times itself three times gives you eight.

The cube of 8 is 2, because 2 times 2 times 2 is 8. Some roots have coefficients, like 4 square roots of 7. The coefficient is the number in front of the radical, and if there isn’t a coefficient shown, like with these numbers the square root of 4, square root of 9, square root of 3, or the cube root of 8, then it’s understood to be 1 since one times any number is itself.

But, this coefficient is just a number that’s multiplied by the radical like 4 times the square root of 7, and it’s written without any multiplication sign which is understood that this is 4 times the square root of 7.

## Practice Questions

**Question #1:**

Which set of numbers contains all perfect squares?

33, 99, 55, 66

33, 99, 55, 66

36, 9, 25, 100

81, 36, 25, 41

**Answer:**

The correct answer is 36, 9, 25, 100. Thirty-six is a perfect square composed of \(6×6\), nine is a perfect square composed of \(3×3\), twenty-five is a perfect square composed of \(5×5\), and one hundred is a perfect square composed of \(10×10\).

**Question #2:**

Which pair shows a correct match between the perfect square and its whole number square root?

**Answer:**

The correct answer is \(\sqrt{144}=12\). One hundred forty-four is a perfect square composed of \(12×12\). Thirty-six is a perfect square, but it is not composed of \(2×2\). It is instead composed of \(6×6\). Twenty-five is a perfect square, but it is not composed of \(4×4\). It is instead composed of \(5×5\). One hundred is a perfect square, but it is not composed of \(11×11\). It is instead composed of \(10×10\).

**Question #3:**

What is the square root of 49?

6

4

5

7

**Answer:**

The correct answer is 7. The square root of 49 is 7, because \(7×7=49\). This also means that 49 is a perfect square.

**Question #4:**

Which value is NOT a perfect square?

81

100

64

99

**Answer:**

The correct answer is 99. Because there is no whole number that can be multiplied by itself to equal 99, it is not a perfect square. Eighty-one is a perfect square composed of \(9×9\), one hundred is a perfect square composed of \(10×10\), and sixty-four is a perfect square composed of \(8×8\).

**Question #5:**

Which pair shows an incorrect match between the perfect square and its whole number square root?

\(\sqrt{49}\) and \(7\)

\(\sqrt{4}\) and \(2\)

\(\sqrt{64}\) and \(8\)

\(\sqrt{9}\) and \(6\)

**Answer:**

The correct answer is \(\sqrt{9}\) and \(6\). The square root of 9 is a perfect square, but it is composed of \(3×3\), not \(6×6\).