Question

Find the value of ${\left( {\dfrac{1}{2}} \right)^{ - 5}}$

Answer 1

Hint: Here we are given a fraction with a negative exponent. An exponent is a number written above and to the right of the mathematical expression called the base. For example in ${a^n} = a \times n{\text{ }}number{\text{ }}of{\text{ }}times$ where $a$ is any base and a non-zero integer and $n$ is exponent or power and any integer. We can solve the given expression by multiplying the numerator and denominator with the given exponent $ - 5$ and thereby opening the bracket.

Complete step by step solution:
Given ${\left( {\dfrac{1}{2}} \right)^{ - 5}}$
When we open the bracket we get $\dfrac{{{1^{ - 5}}}}{{{2^{ - 5}}}}$. Now we have both the numerator and denominator with negative power. To convert the negative powers into positive ones we reciprocate (change the value of numerator with the value of denominator and vice versa) the fraction.
When we reciprocate the fraction we get, $\dfrac{{{2^5}}}{{{1^5}}}$.
Now, we solve numerator and denominator by writing them in expanded form.
$ \Rightarrow \dfrac{{2 \times 2 \times 2 \times 2 \times 2}}{{1 \times 1 \times 1 \times 1 \times 1}}$
$ = \dfrac{{4 \times 4 \times 2}}{1}$
On simplifying, we get
$ = \dfrac{{32}}{1}$
We know that when we multiply $2$ five times and $1$ five times we get $32$ and $1$ respectively.
So, our answer to the given problem is $\dfrac{{32}}{1}$. But we can write $\dfrac{{32}}{1}$ as $32$ as well because 32 divided by $1$ is $32$. So, we get our final answer to the given problem as $32$.
So, the correct answer is “32”.

Note: Remember that an exponent refers to the number of times a number is multiplied by itself. Also when we read such a type of expression we say it as $a$ raised to the power $n$. When an exponent is a negative number the result is always a fraction just like in this question our answer was $\dfrac{{32}}{1}$.