Question

What is $2$ to the power of $-3?$

Answer 1

Hint: To solve this question one should know the concept of exponents. We have to know the laws and properties of exponent to solve this question. We will first write the statement in mathematical form, then apply ${{a}^{-n}}={{\left( \dfrac{1}{a} \right)}^{n}}$ and simplify to reach the final answer.

Complete step by step solution:
The question here is to find the value of $2$ to the power of $-3$, which is mathematically represented as ${{2}^{-3}}$. One of the property that we need to solve this problem is ${{a}^{-n}}={{\left( \dfrac{1}{a} \right)}^{n}}$. This means that if we are asked to find the value of ${{a}^{-n}}$, then the first step would be to make the power positive, by writing the number in its reciprocal form, as done in the next step:
$\Rightarrow {{2}^{-3}}={{\left( \dfrac{1}{2} \right)}^{3}}$
It is a well-known fact that 1 to the power $n$ , where $n$ can be anything gives the value as $1$ itself. So we can perform the next step as below,
$\Rightarrow {{\left( \dfrac{1}{2} \right)}^{3}}=\dfrac{1}{{{2}^{3}}}$
We know that the next step would be to expand the number:
$\Rightarrow \left( \dfrac{1}{2\times 2\times 2} \right)$
On multiplying the denominator terms together in the above question, we get
$\Rightarrow \dfrac{1}{8}$
The question thus results in the value $\dfrac{1}{8}$.

$\therefore $The value of $2$ to the power of $-3$ is $\dfrac{1}{8}$.

Note: We can check the answer whether it is correct or not. This could be done by performing the function from end.
If ${{a}^{n}}=k$, then ${{k}^{\dfrac{1}{n}}}=a$ , applying the same concept to check the answer.
If we evaluate ${{\left( \dfrac{1}{8} \right)}^{-\dfrac{1}{3}}}$ and the result matches the main question, then the answer came is correct.
Now, let's check ${{\left( \dfrac{1}{8} \right)}^{-\dfrac{1}{3}}}$,
As per the property of exponent we can reciprocal the number to change the sign of the power on the number, so ${{\left( \dfrac{1}{8} \right)}^{-\dfrac{1}{3}}}$ could be written as ${{\left( 8 \right)}^{\dfrac{1}{3}}}$. This becomes the question of cube root, so answer which we get is ${{\left( 2\times 2\times 2 \right)}^{\dfrac{1}{3}}}$ , Therefore the base number we get is $2$. This shows that the answer we got is correct.