Question

How do you evaluate \[{\left( {{5^2}} \right)^{ - 3}}\] ?

Answer 1

Hint:We can see this problem is from indices and powers. This number given is having $5$ as base and $2$ as power. But since we have to simplify, we will first evaluate the number inside the bracket and then evaluate the power outside the bracket. We will use the laws of exponents and powers in order to solve the problem.

Complete step by step answer:
So, the given question requires us to simplify \[{5^2}\] to the power $ - 3$. So, we have, \[{\left( {{5^2}} \right)^{ - 3}}\]. This is of the form \[{\left( {{a^m}} \right)^n}\]. But we can rewrite the expression using the laws of indices and powers,
${\left( {{a^m}} \right)^n} = {a^{mn}}$ .
Thus, we will apply the same law on the question above. So, we get,
\[ \Rightarrow {\left( {{5^2}} \right)^{ - 3}} = {5^{2 \times \left( { - 3} \right)}}\]

Simplifying the calculations, we get,
\[ \Rightarrow {\left( {{5^2}} \right)^{ - 3}} = {5^{ - 6}}\]
Now, we will have to deal with the power or the exponent. We know that negative sign in the power means that we have to take reciprocal of the number. So, we get,
\[ \Rightarrow {\left( {{5^2}} \right)^{ - 3}} = \left( {\dfrac{1}{{{5^6}}}} \right)\]
Now, we will calculate the required power of five. So, we know that \[{5^6} = 5 \times 5 \times 5 \times 5 \times 5 \times 5\]. So, we have, \[{5^6} = 15625\].
\[ \therefore {\left( {{5^2}} \right)^{ - 3}} = \left( {\dfrac{1}{{15625}}} \right)\]

Therefore, the value of \[{\left( {{5^2}} \right)^{ - 3}}\] is \[\dfrac{1}{{15625}}\].

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 are fractions with 1 as numerator and respective root in denominator.