Perfect Squares & Square Roots

Q: Why does $\sqrt{64}$ equal $8$ and not $\pm 8$?

By convention, the radical symbol means the positive (principal) root, so $\sqrt{64} = 8$. The two answers $\pm 8$ only appear when you solve an equation like $x^{2} = 64$, because both $8$ and $-8$ square to $64$.

Q: Can you take the square root of a negative number?

Not with the real numbers you use in pre-algebra. No real number times itself gives a negative result — a positive squared is positive, and a negative squared is also positive. So $\sqrt{-25}$ has no real-number answer.

Q: What if the number is not a perfect square?

Then the square root is irrational — a decimal that never ends or repeats. Trap it between the two nearest perfect squares to estimate it: $\sqrt{30}$ sits between $\sqrt{25} = 5$ and $\sqrt{36} = 6$, closer to $5.5$. For exact work you leave it written as $\sqrt{30}$.

The square root of a number asks one question: what number, multiplied by itself, gives me this? Since $7 \cdot 7 = 49$ , the square root of $49$ is $7$ — written $\sqrt{49} = 7$ . Squaring and taking a square root are opposite operations: one builds the square, the other takes it apart.

Square roots show up everywhere in pre-algebra and beyond — finding the side of a square from its area, solving equations like $x^{2} = 64$ , and later the Pythagorean theorem. The whole topic gets much easier once you know the perfect squares up to about $15^{2}$ cold.

Perfect squares are the foundation

A perfect square is what you get when a whole number is multiplied by itself: $1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, \ldots$ The square roots of these numbers come out as whole numbers, which is why they are the easy cases: $\sqrt{144} = 12$ because $12 \cdot 12 = 144$ .

The most common mistake is dividing by $2$ instead of taking a root. $\sqrt{100}$ is $10$ , not $50$ . Halving a number and finding what multiplies by itself to make that number are completely different questions.

Estimating roots that are not perfect

Most numbers are not perfect squares, so their square roots are not whole numbers — $\sqrt{50}$ is an irrational number that goes on forever without repeating. You can still pin it down: find the two perfect squares it sits between. Since $49 < 50 < 64$ , you know $7 < \sqrt{50} < 8$ , and because $50$ is barely past $49$ , the root is just barely past $7$ (it is about $7.07$ ). The number line below shows $\sqrt{50}$ landing in the shaded gap between the perfect-square roots $7$ and $8$ .

This trap-it-between-two-squares move is exactly what tests ask for: which two whole numbers is $\sqrt{50}$ between, and which is it closer to? You never need a calculator for that.

Square roots in equations

Taking a square root is how you solve equations like $x^{2} = 64$ : the value of $x$ is a number that multiplies by itself to give $64$ . There are two such numbers, $8$ and $-8$ , because $(-8)(-8) = 64$ too. So the full solution is $x = 8$ or $x = -8$ , often written $x = \pm 8$ .

One important distinction: the symbol $\sqrt{64}$ by itself means the positive root only, so $\sqrt{64} = 8$ . The $\pm$ only appears when you are solving an equation. And in a geometry problem — a side length or a distance — only the positive root makes sense, so you drop the negative answer.

Worked examples

Example 1: a perfect square

Evaluate $\sqrt{49}$ .

Ask: what number times itself gives

49

7 \cdot 7 = 49

So the square root is

7

\sqrt{49} = 7

Answer: $\sqrt{49} = 7$

Example 2: from area to side length

A square rug covers $144$ square feet. How long is each side?

The area of a square is side times side

s \cdot s = 144

Rewrite with a square

s^{2} = 144

Take the square root

s = \sqrt{144} = 12

A side length must be positive, so keep only

12

Answer: Each side is $12$ feet.

Example 3: estimating a root

Between which two whole numbers is $\sqrt{50}$ , and which is it closer to?

Find the perfect squares on either side of

50

49 < 50 < 64

Take square roots across the inequality

7 < \sqrt{50} < 8

50

is only

1

past

49

but

14

away from

64

, so the root is much closer to

7

Answer: $\sqrt{50}$ is between $7$ and $8$ , closer to $7$ (about $7.07$ ).

Example 4: solving an equation

Solve $x^{2} = 64$ .

Start with the equation

x^{2} = 64

Take the square root of both sides — remember both signs

x = \pm\sqrt{64}

Evaluate

x = 8 \text{ or } x = -8

Check:

8 \cdot 8 = 64

and

(-8)(-8) = 64

✓

Answer: $x = 8$ or $x = -8$

Try one yourself

Common questions

Why does

\sqrt{64}

equal

8

and not

\pm 8

By convention, the radical symbol means the positive (principal) root, so $\sqrt{64} = 8$ . The two answers $\pm 8$ only appear when you solve an equation like $x^{2} = 64$ , because both $8$ and $-8$ square to $64$ .

Can you take the square root of a negative number?

Not with the real numbers you use in pre-algebra. No real number times itself gives a negative result — a positive squared is positive, and a negative squared is also positive. So $\sqrt{-25}$ has no real-number answer.

What if the number is not a perfect square?

Then the square root is irrational — a decimal that never ends or repeats. Trap it between the two nearest perfect squares to estimate it: $\sqrt{30}$ sits between $\sqrt{25} = 5$ and $\sqrt{36} = 6$ , closer to $5.5$ . For exact work you leave it written as $\sqrt{30}$ .

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