Question

How do you simplify \[\sqrt{{{x}^{15}}}\]?

Answer 1

Hint: In this problem, we have to simplify the given square root. We can separate the terms inside the square root using the product of exponent rule. By using the product of exponent, we can convert into two terms. We can then take individual roots using the multiplication of roots and we can get the simplified value.

Complete step by step answer:
We know that the given square root to be simplified is,
\[\sqrt{{{x}^{15}}}\]
We can now use the product of exponent rule.
We know that the product of exponent rule is,
\[{{x}^{n+m}}={{x}^{n}}\times {{x}^{m}}\]
We can use this in the given square root to separate the terms inside the square root, we get
\[\Rightarrow \sqrt{{{x}^{14}}\times {{x}^{1}}}\]
We can now take individual roots for the terms inside the square root using the multiplication of roots.
We know that the multiplication of roots formula is,
\[\Rightarrow \sqrt{xy}=\sqrt{x}\times \sqrt{y}\]
We can apply this in the above step to take individual roots for the terms inside the square root using the multiplication of roots.
\[\Rightarrow \sqrt{{{x}^{14}}}\times \sqrt{x}\]
We can now write the above step as,
\[\Rightarrow \sqrt{{{\left( {{x}^{7}} \right)}^{2}}}\times \sqrt{x}\]
Now we can cancel the square and the square root in the left-hand side, we get
\[\Rightarrow {{x}^{7}}\sqrt{x}\]
Therefore, the simplified form of the given square root \[\sqrt{{{x}^{15}}}\] is \[{{x}^{7}}\sqrt{x}\].

Note:
We should remember the formula for the product rule of exponents and the multiplication of roots formula to be applied for these types of problems. We should also remember that we can cancel the square and the square root.