Squares and Square Roots

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: $$$\color{purple}{a^b=\underbrace{a\cdot a\cdot a\cdot a\cdot...\cdot a}_{b}}$$$.

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): $$${{\left(-{2}\right)}}^{{2}}={4}$$$, $$${{\left(\frac{{3}}{{4}}\right)}}^{{2}}=\frac{{9}}{{16}}$$$, $$${{\left(-{0.3}\right)}}^{{2}}={0.09}$$$.

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.

The square root of the number has a special notation: $$$\color{purple}{\sqrt{b}}$$$ (the plain text is 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:

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.

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.

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