Question

Find the value of \[{125^{ - \dfrac{2}{3}}}\].

Answer 1

Hint:Since, the exponent of the number is in negative form and we have according to the law of indices \[{a^{ - m}} = \dfrac{1}{{{a^m}}}\]. So, we have only one option to get rid of the exponent that is to get the reciprocal of the number and exponent together. First 3 on the bottom of the fraction and cube root of 125 should be handled.

Complete step by step solution:
Law of indices states that the denominator of the fraction is the root of the number or letter, and that the numerator of the fraction is the power to raise the answer to.
According to the law of indices we have,
\[{a^{ - m}} = \dfrac{1}{{{a^m}}}\]
So, taking reciprocal of \[{125^{ - \dfrac{2}{3}}}\]we have,
\[\dfrac{1}{{{{125}^{\dfrac{2}{3}}}}}\]
When the exponent of the numerator goes to the denominator then the sign changes that is why we have a positive exponent in the denominator.
Now solving we have,
\[ \Rightarrow \dfrac{1}{{{{\left( {{{125}^{\dfrac{1}{3}}}} \right)}^2}}}\]
The exponent \[\frac{1}{3}\] when is taken out then it will be the cube root of \[125\]which is 5. Hence we will have only 5 and its square in the denominator.
\[ \Rightarrow \dfrac{1}{{{{\left( 5 \right)}^2}}}\]
On squaring the denominator we have,
\[ \Rightarrow \dfrac{1}{{25}}\]
Therefore, the value of \[{125^{ - \dfrac{2}{3}}}\]is\[\dfrac{1}{{25}}\].

Note: Whenever there are exponents then in such cases it is advised to take the reciprocal which will convert the problem into a simpler form. The exponents which are in the form of fraction can be taken as the root of the number and that root will be the interchanged that is the numerator will be interchanged with the denominator. When the exponent of numerator goes to the denominator or vice versa then always there is change in sign.