Question

How do you simplify ${\left( { - 8} \right)^{\dfrac{5}{3}}}$ ?

Answer 1

Hint:We can see this problem is from indices and powers. This number given is having $\left( { - 8} \right)$ as base and $\left( {\dfrac{5}{3}} \right)$ as power. We will first express our base number $8$ 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 answer:
So, we have, ${\left( { - 8} \right)^{\dfrac{5}{3}}}$.
So, we know the factorisation of $8$ is $8 = 2 \times 2 \times 2$.
Expressing in exponential form, we get, $8 = {2^3}$.
So, we get, ${\left( { - 8} \right)^{\dfrac{5}{3}}} = {\left( { - {2^3}} \right)^{\dfrac{5}{3}}}$.
Using the law of exponents and powers ${\left( {{a^x}} \right)^y} = \left( {{a^{xy}}} \right)$, we get,
 $ \Rightarrow {\left( { - 8} \right)^{\dfrac{5}{3}}} = \left( { - {2^{3 \times \dfrac{5}{3}}}} \right)$
Simplifying the power of two, we get,
$ \Rightarrow {\left( { - 8} \right)^{\dfrac{5}{3}}} = \left( { - {2^5}} \right)$
Taking the negative sign outside of the bracket,
$ \Rightarrow {\left( { - 8} \right)^{\dfrac{5}{3}}} = - \left( {{2^5}} \right)$
Evaluating the value of expression, we get,
$ \therefore {\left( { - 8} \right)^{\dfrac{5}{3}}} = - 32$

Therefore, the value of ${\left( { - 8} \right)^{\dfrac{5}{3}}}$ 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. Negative power of a number means that we have to take the reciprocal of the expression. Fractional powers like half or one-third indicate that we have to take the square root or cube root of the number.