Squaring multiplies a number by itself: $7^{2} = 7 \cdot 7 = 49$ (see whole-number exponents). A square root goes the other way.
Why it works
A square with sides 7 units long has area $7 \times 7 = 49$ square units. So if a square’s area is 49, its side must be 7. That is what a square root asks: “What number times itself makes 49?”
Key idea
$\sqrt{n}$ is the number, not negative, whose square is $n$. The perfect squares $1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, \ldots$ have whole-number square roots.
Negative numbers have squares too: $(-7)^{2} = 49$. The symbol $\sqrt{\ }$ always means the positive root, and we write $-\sqrt{49} = -7$ for the negative one.
Worked examples
Worked example: A perfect square
Find $\sqrt{81}$.
- Which number times itself is 81?
- $9 \times 9 = 81$.
Answer: $9$
Worked example: A fraction
Find $\sqrt{\dfrac{16}{25}}$.
- Try the root of the top over the root of the bottom: $\frac{4}{5}$.
- Check: $\frac{4}{5} \cdot \frac{4}{5} = \frac{16}{25}$.
Answer: $\frac{4}{5}$
Worked example: A decimal
Find $\sqrt{0.36}$.
- $0.36 = \frac{36}{100}$, and $\sqrt{36} = 6$, $\sqrt{100} = 10$.
- So the root is $\frac{6}{10} = 0.6$. Check: $0.6 \times 0.6 = 0.36$.
Answer: $0.6$
Common mistake
Halving the number: $\sqrt{16} = 8$. Check by squaring your answer: $8^{2} = 64$, not 16. The root of 16 is 4, because $4 \cdot 4 = 16$.