Question

How do you simplify ${16^{\dfrac{5}{4}}}$?

Answer 1

Hint: In order to write the expression into the simplest form, factorize the base part of the value . Such that it contains the number which is raised to the power of 4 , so that we can cancel out in it. Since in our case we have the largest factor as 16 who is a number which is raised to the power of 4. So we’ll use here prime factorisation of 16 and pull out terms and write it as power of 4 .

Complete step-by-step solution:
Given a expression to simplify ,
${16^{\dfrac{5}{4}}}$
First Separating the value $16$ into its factors, So the factors of $16$ comes to be,
$4and4$
But we need to find the factors , such that it contains the number which is raised to the power of 4 ,
 We will further do prime factorisation of $4and4$,
$2,2,2and2$.
Now we can write it as ${2^4}$ instead of $16$ so that we can simplify the question consider the cancellation of the exponent having denominator 4 with the form of the factors of $16$ and , we get
=$
  {16^{\dfrac{5}{4}}} \\
  {({2^4})^{\dfrac{5}{4}}} \\
    \\
 $
From the above , we can say that $16 = {2^4}$
Replace $16$ as ${2^4}$ in the original number and
Taking out $
  {2^{4 \times \dfrac{5}{4}}} \\
    \\
 $ to simplify it ..
=${2^{4 \times \dfrac{5}{4}}}$
Therefore , we are left with ${2^5}$ in the simplest form, which gives the single answer as $32$.
Formula:
$
  {\left( a \right)^{\dfrac{m}{n}}} = {\left( {{a^m}} \right)^{\dfrac{1}{n}}} \\
  {a^{m + n}} = {a^m} \times {a^n} \\
 $
Doing Prime factorisation of a number helps a lot in simplification.

Note:
1. Make sure the calculation in the question is done correctly.
2.To calculate the simplified answer try to break out the steps from the question.
3.Always check the required formula exponent rule and try to cancel out the common factors.
4. If the base of the exponent number is prime, we cannot simplify the question further and answer is obtained by simply calculating the exponent value.