Question

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

Answer 1

Hint: To simplify the above expression given in the above problem i.e. ${{16}^{-\dfrac{3}{4}}}$, we are going to write 16 as ${{2}^{4}}$. You might be thinking why we are writing 16 as ${{2}^{4}}$ because then the exponent of 16 which is having 4 in the denominator will be cancelled out and we can easily simplify the given expression. Also, we are going to use the property of exponent which states that when the power of any number is negative then we have to take the reciprocal of the number to eliminate that negative power.

Complete step by step solution:
The expression given above which we are going to simplify is as follows:
${{16}^{-\dfrac{3}{4}}}$
In the above expression, we can write 16 as ${{2}^{4}}$ then the above expression will look like:
$={{\left( {{2}^{4}} \right)}^{-\dfrac{3}{4}}}$
Now, we know the property of the exponents of a number which states that:
${{\left( {{a}^{b}} \right)}^{c}}={{a}^{bc}}$
Then we can multiply the exponents of the number 2 and then the expression will look as follows:
$\Rightarrow {{2}^{4\times -\dfrac{3}{4}}}$
Now, 4 will be cancelled out from the numerator and the denominator in the exponent and we get,
$={{2}^{-3}}$
Now, rearranging the above number we get,
$={{\left( {{2}^{3}} \right)}^{-1}}$
In the above expression, we are going to see the power of 2 as 3 which means that we have to multiply 2 by 3 times then we get,
$\begin{align}
  & ={{\left( 2\times 2\times 2 \right)}^{-1}} \\
 & ={{8}^{-1}} \\
\end{align}$
We know that when the exponent of any number is -1 then we can eliminate the exponent -1 by taking the reciprocal of that number so we are going to take the reciprocal of 8 so that -1 in the exponent will get eliminated.
$\Rightarrow \dfrac{1}{8}$
Hence, we have simplified the above expression to $\dfrac{1}{8}$.

Note: The trick which we have learnt from this problem is that whenever you see any fraction in the exponent of a number then try to convert the number into the same number which is given in the denominator of the exponent of that fraction. As denominator of the fraction written in the exponent of the given number 16 is 4 so we have converted the number 16 into 2 to the power of 4.