Question

How do you simplify $${64}^{-{\displaystyle \frac{1}{3}}}$$?

Answer 1

Hint: Here, we will rewrite 64 as power of some number. Then substitute this value in the given expression and we will then use the suitable exponential formula to simplify the expression. We will then apply the negative exponent rule and simplify it to get the required answer.

Formula used:
 According to the property of the exponential function, the negative exponent rule is given by $$a^{-1}={\displaystyle \frac{1}{a}}$$.

Complete step by step solution:
Here we need to simplify the given expression. The given expression is $${64}^{-{\displaystyle \frac{1}{3}}}$$.
Now, we will write the base of the exponential i.e. 64 in the power form.
We know that when we multiply the exponents with the same base, then their power gets added.
We can write 64 as
$$64=4^3$$
Now, we will put this value in the expression i.e. in $${64}^{-{\displaystyle \frac{1}{3}}}$$. Therefore, we get
$${64}^{-{\displaystyle \frac{1}{3}}}={\left(4^3\right)}^{-{\displaystyle \frac{1}{3}}}$$
We know that when we take the power of the exponentials, then the powers of exponents get multiplied.
So, using the formula $${\left(a^m\right)}^n=a^{m\times n}$$, we get
$$\Rightarrow {64}^{-{\displaystyle \frac{1}{3}}}={4^{3\times }}^{{\displaystyle \frac{-1}{3}}}$$
On multiplying the powers, we get
$$\Rightarrow {64}^{-{\displaystyle \frac{1}{3}}}=4^{-1}$$
We know from the property of the exponentials that $$a^{-1}={\displaystyle \frac{1}{a}}$$.
Using this property of exponents, we get

$$\Rightarrow {64}^{-{\displaystyle \frac{1}{3}}}={\displaystyle \frac{1}{4}}$$
Hence, this is the required simplified value of the given expression.

Note:
An expression that represents the repeated multiplication of the same number is known as power. Whereas, when a number is written with power then the power becomes the exponent of that particular number. It shows the number of times that particular number will be multiplied by itself. Hence, whenever we are given the multiplication of the same numbers then, we can express that number with an exponent.