Question

The value of $${\left( {\dfrac{{ - 1}}{{216}}} \right)^{\dfrac{{ - 2}}{3}}}$$
(A) $$36$$
(B) $$ - 36$$
(C) $$\dfrac{1}{{36}}$$
(D) $$\dfrac{{ - 1}}{{36}}$$

Answer 1

Hint: The given question involves powers. And the base in the form of a fraction. The denominator is a perfect cube, so we use that and simplify the base. The, we multiply the powers. We will be left with one negative power. So, we take the reciprocal of the base and get to the final answer by multiplying the power with the base.
The formulae used to solve the problem are as follows:
$$ - \times - = + $$
$${\left( {\dfrac{a}{b}} \right)^m} = \left( {\dfrac{{{a^m}}}{{{b^m}}}} \right)$$
$${\left( {{{\left( {\dfrac{a}{b}} \right)}^m}} \right)^n} = {\left( {\dfrac{a}{b}} \right)^{m \times n}}$$
$${a^{ - m}} = \dfrac{1}{{{a^m}}}$$

Complete step-by-step solution:
Let us consider the given expression,
$${\left( {\dfrac{{ - 1}}{{216}}} \right)^{\dfrac{{ - 2}}{3}}}$$
We know that the denominator, $$216$$ can be expressed as cube.
So, we can write the above expression as,
$$= {\left( {\dfrac{{ - 1}}{{{{\left( 6 \right)}^3}}}} \right)^{\dfrac{{ - 2}}{3}}}$$
Since numerator is $$1$$ we can cube that as well,
$$= {\left( {\dfrac{{{{\left( { - 1} \right)}^3}}}{{{{\left( 6 \right)}^3}}}} \right)^{\dfrac{{ - 2}}{3}}}$$
We know that,
$$\left( {\dfrac{{{a^m}}}{{{b^m}}}} \right) = {\left( {\dfrac{a}{b}} \right)^m}$$
So, we can now take the power common to both the numerator and denominator,
$$ = {\left( {{{\left( {\dfrac{{ - 1}}{6}} \right)}^3}} \right)^{\dfrac{{ - 2}}{3}}}$$
This is of the form, $${\left( {{{\left( {\dfrac{a}{b}} \right)}^m}} \right)^n}$$
We have, $${\left( {{{\left( {\dfrac{a}{b}} \right)}^m}} \right)^n} = {\left( {\dfrac{a}{b}} \right)^{m \times n}}$$
So, the above expression becomes,
$$ ={\left( {\dfrac{{ - 1}}{6}} \right)^{3 \times \dfrac{{ - 2}}{3}}}$$
The $$3$$ in the numerator and the $$3$$in the denominator gets cancelled,
We now have,
$$= {\left( {\dfrac{{ - 1}}{6}} \right)^{ - 2}}$$
If the power is negative, we have, $${a^{ - m}} = \dfrac{1}{{{a^m}}}$$
Since the power is negative, base takes the reciprocal,
$$= {\left( { - 6} \right)^2}$$
Now we square the given term,
We know that $$ - \times - = + $$
$$= - 6 \times - 6 = 36$$
Therefore, the final answer is $$36$$
Hence, option (A) is the correct answer.

Note: Since the base is a negative fraction and the power is also negative, we need to be careful while solving the problem. Remember all the power and base related formulae. If there are any perfect squares or cubes in the question, make sure that you convert it first. This will help in cancelling out the powers.