Question

How do you simplify: $\sqrt{{{x}^{16}}}$ ?

Answer 1

Hint: We are given $\sqrt{{{x}^{16}}}$ , we are asked to simplify, to find the simplified value of $\sqrt{{{x}^{16}}}$ , we will learn how the value are taken out of the square root, we then work on factoring the ${{x}^{16}}$ , we will make pair of 20- term, and in place of pair of 1-pair just 1 term will came out of the square root and solve our problem.

Complete step by step answer:
We are given $\sqrt{{{x}^{16}}}$, we have to simplify the term inside the square root, to simplify the value we should know that only the term which are in pair of 2 can come out and when we take them out 2 term become one.
For example: say we have $\sqrt{9}$ , as $9=3\times 3$ so $\sqrt{9}=\sqrt{3\times 3}$ , here we have pair of 3, so 3 will come out, 3 will become when taken out of root.
So, $\sqrt{9}=3$ .
If we have a term whose pair is not made, then that term will stay inside square root only.
For example:
$\sqrt{12}$ as $12=2\times 2\times 2$ so,
$\Rightarrow \sqrt{12}=\sqrt{2\times 2\times 3}$ .
Here we have a pair of 2, and 3 is not in pair, so 2 will come out while 3 remains inside it.
So, $\Rightarrow \sqrt{12}=2\sqrt{3}$
Now we will work on our problem. We have –
$\Rightarrow \sqrt{{{x}^{16}}}$
We factor ${{x}^{16}}$ , so we will get ${{x}^{16}}$ made by product of x, 16 times.
So, ${{x}^{16}}$ is a product of 16 x.
When we make a pair of 2-2, we will be getting 8 pairs as 16 will be divided into 8 pairs of 2. So,
$\Rightarrow \sqrt{{{x}^{16}}}=\sqrt{x\times x\times x\times x\times x\times x\times x\times x\times x\times x\times x\times x\times x\times x\times x\times x}$
So, when we simplify, 8 x will come out of the square root. So, we get –
$\Rightarrow \sqrt{{{x}^{16}}}=x\times x\times x\times x\times x\times x\times x\times x$
Now as $x\times x={{x}^{2}}$ , $x\times x\times x={{x}^{3}}$ so multiplying x eight times will give us ${{x}^{8}}$ .
Hence, $\sqrt{{{x}^{16}}}={{x}^{8}}$ .
So, we get a simplified value of $\sqrt{{{x}^{16}}}$ is ${{x}^{8}}$ .

Note:
Another way to solve this is to use the surds.
We know that $\sqrt{x}={{\left( x \right)}^{{}^{1}/{}_{2}}}$
So, $\Rightarrow \sqrt{{{x}^{16}}}={{\left( {{x}^{16}} \right)}^{{}^{1}/{}_{2}}}$ .
Now as we know that ${{\left( {{x}^{a}} \right)}^{b}}={{x}^{a\times b}}$ .
So, in our above equation we have ${{\left( {{x}^{16}} \right)}^{{}^{1}/{}_{2}}}$ .
So, using identity ${{\left( {{x}^{a}} \right)}^{b}}={{x}^{ab}}$ -
We get –
$\Rightarrow {{\left( {{x}^{16}} \right)}^{{}^{1}/{}_{2}}}={{x}^{16\times {}^{1}/{}_{2}}}$
By simplifying, we get –
$={{x}^{{}^{16}/{}_{2}}}$
By solving, we get –
$\dfrac{16}{2}=8$ so,
$={{x}^{8}}$
Hence we get –
$\sqrt{{{x}^{16}}}={{x}^{8}}$ .