Question

Simplify the value of ${\left( {16} \right)^{\dfrac{5}{4}}}$.

Answer 1

Hint: We can see this problem is from indices and powers. This number given has $16$ as base and $\left( {\dfrac{5}{4}} \right)$ as power. We will first express our base number $16$ in powers of two. Then, we will simplify the expression using the law of exponents and powers ${\left( {{a^x}} \right)^y} = \left( {{a^{xy}}} \right)$. Then, we will find the resultant expression by evaluating the final power of two.

Complete step by step solution:
We have, ${\left( {16} \right)^{\dfrac{5}{4}}}$.
We know the factorisation of $16$ is $16 = 2 \times 2 \times 2 \times 2$.
Expressing it in exponential form, we get, $16 = {2^4}$.
So, we get, ${\left( {16} \right)^{\dfrac{5}{4}}} = {\left( {{2^4}} \right)^{\dfrac{5}{4}}}$.
Using the law of exponents and powers ${\left( {{a^x}} \right)^y} = \left( {{a^{xy}}} \right)$, we get,
 $ \Rightarrow {\left( {16} \right)^{\dfrac{5}{4}}} = {2^{4 \times \dfrac{5}{4}}}$
Simplifying the power of two, we get,
$ \Rightarrow {\left( {16} \right)^{\dfrac{5}{4}}} = {2^5}$
Simplifying the expression,
$ \Rightarrow {\left( {16} \right)^{\dfrac{5}{4}}} = {2^5}$
We know that the fifth power of two is $32$.
Evaluating the value of expression, we get,
$ \Rightarrow {\left( {16} \right)^{\dfrac{5}{4}}} = 32$
Therefore, the value of ${\left( {16} \right)^{\dfrac{5}{4}}}$ is $32$.

Note:
These rules or laws of indices help us to minimize the problems and get the answer in very less time. These powers can be positive and negative but can be moulded according to our convenience while solving the problem. Also note that cube-root, square-root is fractions with 1 as numerator and respective root in the denominator.