Question

Find the value of ${{\left( -{{5}^{5}} \right)}^{3}}$.
(a) ${{\left( -5 \right)}^{8}}$
(b) ${{\left( -5 \right)}^{15}}$
(c) ${{\left( -5 \right)}^{10}}$
(d) ${{5}^{15}}$

Answer 1

Hint: First, before proceeding for this , we must know the following rules of the exponents to get the desired result as ${{\left( {{a}^{m}} \right)}^{n}}={{a}^{m\times n}}$. Then, to get the final result of the following question, we must use the above formula to get the final result of the exponents. Then, by applying the rules stated above, we can find the value of the given exponent in the question.

Complete step-by-step answer:
In this question, we are supposed to find the value of the exponents.
So, before proceeding for this , we must know the following rules of the exponents to get the desired result as:
${{\left( {{a}^{m}} \right)}^{n}}={{a}^{m\times n}}$
So, to get the final result of the following question, we must use the above formula to get the final result of the exponents.
But, before proceeding we must know the fact that a negative number will remain negative if its power is an odd number.
Oppositely, a negative number will become positive if its power is an even number.
Now, by applying the rules stated above, we can find the value of the given exponent in the question as:
${{\left( -{{5}^{5}} \right)}^{3}}={{\left( -5 \right)}^{5\times 3}}$
Now, by multiplying the terms in the power to get the final result of the given question.
So, the further proceeding will be as follows:
${{\left( -5 \right)}^{5\times 3}}={{\left( -5 \right)}^{15}}$
So, the final result of the given expression in question is ${{\left( -5 \right)}^{15}}$.
Hence, from the given options, option (b) is correct.

Note: Now, to solve these types of questions we need to know some of the basic rules of the exponents beforehand so that we can easily proceed in these types of questions. Then, some of the basic rules are:
$\begin{align}
  & {{\left( {{a}^{m}} \right)}^{n}}={{a}^{m\times n}} \\
 & {{a}^{m}}\times {{a}^{n}}={{a}^{m+n}} \\
 & \dfrac{{{a}^{m}}}{{{a}^{n}}}={{a}^{m-n}} \\
\end{align}$