Question

How do you simplify $ {\dfrac{1}{{32}}^{\dfrac{{ - 2}}{5}}} $ ?

Answer 1

Hint: We know that the above given number $ {\dfrac{1}{{32}}^{\dfrac{{ - 2}}{5}}} $ is in exponential form. An exponent refers to the number of times a number is multiplied by itself. There is base and exponent or power in this type of equation. Here, in the given question $ \dfrac{1}{{32}} $ is the base and the number $ \dfrac{{ - 2}}{5} $ is the exponential power. As we know that as per the negative exponent rule if there is $ {a^{ - n}} $ then it will change into $ \dfrac{1}{{{a^n}}} $ as ‘n’ is negative. When we express a number in exponential form then we can say that it’s power has been raised by the exponent.

Complete step by step solution:
We can simplify this by using the negative exponent rule $ {a^{ - n}} = \dfrac{1}{{{a^n}}} $ . To solve exponential equations with base, use the property of power of exponential functions.
i.e. $ {\dfrac{1}{{32}}^{\dfrac{{ - 2}}{5}}} = {\left( {{{32}^{ - 1}}} \right)^{\dfrac{{ - 2}}{5}}} $ , So the negative sign gets cancelled i.e. new expression is $ {32^{\dfrac{2}{5}}} $ and we know that $ {(2)^5} = 32 $ so here it will be $ {({2^5})^{\dfrac{2}{5}}} $ .
As we know that another rule of exponent says that $ {({a^m})^p} = {a^{m*p}} $ , so by applying the formula of exponential power equations, here it will be $ {2^{5 \times \dfrac{2}{5}}} = {2^2} $ .
So as the powers of the base multiplied and gave the value $ 4 $ .
Hence the answer of $ {\dfrac{1}{{32}}^{\dfrac{{ - 2}}{5}}} $ is $ 4. $
So, the correct answer is “ $ {\dfrac{1}{{32}}^{\dfrac{{ - 2}}{5}}} $ is $ 4. $ ”.

Note: We know that exponential equations are equations in which variables occur as exponents. We should solve this kind of problem by using the properties of exponents to simplify the problem. We have to keep in mind that if there is a negative value in the power or exponent then it will reverse the number .i.e. $ {m^{ - x}} $ will always be equal to $ \dfrac{1}{{{m^x}}} $ .