Question

How do you simplify \[{16^{ - \dfrac{3}{4}}}\]?

Answer 1

Hint: In the given question, we have been given an expression. This expression contains a number. This number has power. We have to simplify the expression and find its value. To do that, we are first going to simplify the expression using the appropriate identities and then calculate its value.

Formula Used:
In this question we are going to use the formula of index, which is:
\[{a^{\dfrac{m}{n}}} = {\left( {{a^{\dfrac{1}{n}}}} \right)^m}\]

Complete step by step answer:
The given expression is \[{16^{ - \dfrac{3}{4}}}\].
First, we are going to remove the negative power by taking the reciprocal, so,
\[{16^{ - \dfrac{3}{4}}} = \dfrac{1}{{{{16}^{\dfrac{3}{4}}}}}\]
There are two powers –fourth root and cube. First, we are going to find its root, which is:
\[{\left( {16} \right)^{\dfrac{1}{4}}} = 2\]
Now, we are going to calculate the cube of \[2\], which is:
\[{2^3} = 8\]

Hence, \[{16^{ - \dfrac{3}{4}}} = \dfrac{1}{{{{16}^{\dfrac{3}{4}}}}} = \dfrac{1}{8}\]

Additional Information:
In this question, we first found the fourth root of \[16\] and then calculated the cube. We did this because it is easy to calculate the value of a smaller number. But we could have gone the other way around, though it is comparatively lengthy, i.e., first calculate the cube of \[16\], which is \[4096\], and then calculate the fourth root of \[4096\], which is \[8\].

Note:
So, in this question, we first converted the given expression into its smaller form and eliminated the exponent. Then, when we got the most simplified form, i.e., the point from where it cannot get simpler, we applied the remaining index and calculated the value. But to do all that, it was required that we know the formula and know how to apply it.