Question

Simplify the value of: ${\left( {1024} \right)^{ - \dfrac{4}{5}}}$

Answer 1

Hint: We can see this problem is from indices and powers. This number given is having $1024$ as base and $\left( { - \dfrac{4}{5}} \right)$ as power. We will first express our base number $1024$ 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, ${\left( {1024} \right)^{ - \dfrac{4}{5}}}$.
So, we know the factorisation of $\left( {1024} \right)$ is $1024 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2$.
Expressing in exponential form, we get, $1024 = {2^{10}}$.
So, we get, ${\left( {1024} \right)^{ - \dfrac{4}{5}}} = {\left( {{2^{10}}} \right)^{ - \dfrac{4}{5}}}$.
Using the law of exponents and powers ${\left( {{a^x}} \right)^y} = \left( {{a^{xy}}} \right)$, we get,
 $ \Rightarrow {\left( {1024} \right)^{ - \dfrac{4}{5}}} = {\left( 2 \right)^{10 \times \left( { - \dfrac{4}{5}} \right)}}$
Simplifying the power of two, we get,
$ \Rightarrow {\left( {1024} \right)^{ - \dfrac{4}{5}}} = {\left( 2 \right)^{2 \times \left( { - 4} \right)}}$
Simplifying the expression,
$ \Rightarrow {\left( {1024} \right)^{ - \dfrac{4}{5}}} = {\left( 2 \right)^{ - 8}}$
We know that negative power means that we have to take reciprocal of the number. So, we get,
\[ \Rightarrow {\left( {1024} \right)^{ - \dfrac{4}{5}}} = \dfrac{1}{{{2^8}}}\]
Evaluating the value of expression, we get,
\[ \Rightarrow {\left( {1024} \right)^{ - \dfrac{4}{5}}} = \dfrac{1}{{256}}\]
So, the value of ${\left( {1024} \right)^{ - \dfrac{4}{5}}}$ is \[\dfrac{1}{{256}}\].

Note:
The powers or exponents can be positive and negative but can be moulded according to our convenience while solving the problem. Also note that cube-root, square-root are fractions with 1 as numerator and respective root in denominator. Negative power means that we have to take the reciprocal of the number. Fractional power such as ${a^{\dfrac{1}{n}}}$ means that we have to take the nth root of the number a.