# Squares and Square Roots

## 関連する計算機: 平方根計算機

To **square a number**, multiply it by itself.

For, example, square of $$$5$$$ is $$${5}\times{5}={25}$$$.

When we talked about exponents and integers, we said that number $$$a$$$ raised to $$$b$$$-th power is number $$$a$$$ multiplied by itself $$$b$$$ times: $$$\color{purple}{a^b=\underbrace{a\cdot a\cdot a\cdot a\cdot...\cdot a}_{b}}$$$.

So, to square means to **raise to the second power.**

Thus, we can write, that the square of the number $$$5$$$ is

$$$5^{\color{purple}{2}}=25$$$

It is read as "5 squared equals 25".

We can square any numbers: negative numbers, fractions, decimals, etc.

Just a couple of fast examples, because we already did this (see exponents and integers, negative exponents): $$${{\left(-{2}\right)}}^{{2}}={4}$$$, $$${{\left(\frac{{3}}{{4}}\right)}}^{{2}}=\frac{{9}}{{16}}$$$, $$${{\left(-{0.3}\right)}}^{{2}}={0.09}$$$.

Remember, that raising to the power is just multiplying certain number of times, and squaring is raising to the second power, i.e. multiplying number by itself.

Now, suppose we want to do reverse operation. We are given a number, and want to find a value, that when squared will give an initial number. This value is called square root of the number.

**Square root of the number** $$${b}$$$ is such number $$${a}$$$, that $$${{a}}^{{2}}={b}$$$.

The square root of the number has a special notation: $$$\color{purple}{\sqrt{b}}$$$.

For, example since we know that $$${5}\times{5}={25}$$$, then square root of $$$25$$$ is $$$5$$$: $$$\sqrt{{{25}}}=5$$$.

Following table contains squares and squares roots of first 15 whole numbers:

Number $$${a}$$$ (square root) | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 |

Number $$${b}$$$ (square of number) | 0 | 1 | 4 | 9 | 16 | 25 | 36 | 49 | 64 | 81 | 100 | 121 | 144 | 169 | 196 | 225 |

Highlighted numbers are the numbers from above example.

Looking at this, we can easily see, that $$${{11}}^{{2}}={121}$$$ and $$$\sqrt{{{121}}}={11}$$$.

Also, it is worth noting, that these numbers are written on the main diagonal of the multiplication table.

**Square root symbol** $$$\color{purple}{\sqrt{\phantom{0}}}$$$ is also called the **radical.**

As you could notice, any number squared will result in a positive number.

Why is that? Because we always multiply numbers with same signs, and this will result into a positive number.

As a result, **square root of the negative number doesn't exist** (at least for now:)).

Finally, notice that square root undoes squaring (if we take square root of a number and then square the result, we will get back to the original number), and vice versa (with a slight modification):

$$${\color{red}{{{{\left(\sqrt{{{b}}}\right)}}^{{2}}={b}}}}$$$

$$${\color{green}{{\sqrt{{{{b}}^{{2}}}}={\left|{b}\right|}}}}$$$

Did you notice absolute value? Why is that?

Because squaring any number will give a positive number, and taking square root will also give positive number. Absolute value guarantees, that we wil have positive number.

We could write, that $$${{\left(\sqrt{{{b}}}\right)}}^{{2}}={\left|{b}\right|}$$$, but that is not necessary, since $$${b}$$$ is already non-negative number (square root of negative number doesn't exist).

Let's go through a couple of example.

**Example 1.** $$${{\left(\sqrt{{{9}}}\right)}}^{{2}}={{3}}^{{2}}={9}$$$.

**Example 2.** $$${{\left(\sqrt{{{2}}}\right)}}^{{2}}={2}$$$.

**Example 3.** $$${{\left(\sqrt{{-{5}}}\right)}}^{{2}}$$$ doesn't exist.

**Example 4.** $$$\sqrt{{{{0.9}}^{{2}}}}=\sqrt{{{0.81}}}={0.9}$$$.

**Example 5.** $$$\sqrt{{{{\left({\color{red}{{-{4}}}}\right)}}^{{2}}}}=\sqrt{{{16}}}={4}={\left|{\color{red}{{-{4}}}}\right|}$$$.