Question

How do you simplify $ {\left( {\dfrac{2}{5}} \right)^{ - 3}} $ ?

Answer 1

Hint: In this question we need to simplify $ {\left( {\dfrac{2}{5}} \right)^{ - 3}} $ . Here, we know that a negative exponent means the reciprocal of the term. Thus, we will reciprocal the given term and determine the cubes of $ 2 $ and $ 5 $ . And, substitute the values by which we will get the required answer.

Complete step by step answer:
Here, we need to simplify $ {\left( {\dfrac{2}{5}} \right)^{ - 3}} $ .
Now, we have a negative exponent in the given term $ {\left( {\dfrac{2}{5}} \right)^{ - 3}} $ . We know that the negative sign on an exponent means the reciprocal of the given term.
Thus we can rewrite $ {\left( {\dfrac{2}{5}} \right)^{ - 3}} $ as $ \dfrac{1}{{{{\left( {\dfrac{2}{5}} \right)}^3}}} $ .
Therefore, $ {\left( {\dfrac{2}{5}} \right)^{ - 3}} = \dfrac{1}{{{{\left( {\dfrac{2}{5}} \right)}^3}}} $
 $ {\left( {\dfrac{2}{5}} \right)^{ - 3}} = \dfrac{1}{{\dfrac{{{2^3}}}{{{5^3}}}}} $
Here, the denominator is in the faction, thus we can rewrite the term as,
 $ {\left( {\dfrac{2}{5}} \right)^{ - 3}} = \left( {\dfrac{1}{{{2^3}}}} \right) \times \left( {\dfrac{{{5^3}}}{1}} \right) $
  $ {\left( {\dfrac{2}{5}} \right)^{ - 3}} = \left( {\dfrac{{{5^3}}}{{{2^3}}}} \right) $
Now, let us determine the cubes of $ 2 $ and $ 5 $ .
 $ {2^3} = 2 \times 2 \times 2 $
 $ {2^3} = 8 $
Similarly, $ {5^3} = 5 \times 5 \times 5 $
 $ {5^3} = 125 $
Therefore, substituting the values of $ {2^3} $ and $ {5^3} $ , we have,
 $ {\left( {\dfrac{2}{5}} \right)^{ - 3}} = \dfrac{{125}}{8} $
 $ {\left( {\dfrac{2}{5}} \right)^{ - 3}} = 15.625 $
$\Rightarrow$ Hence, the value of $ {\left( {\dfrac{2}{5}} \right)^{ - 3}} $ by simplifying is $ \dfrac{{125}}{8} $ or $ 15.625 $ .

Note:
In this question, it is important to note here that mostly we would come across the positive exponent. The positive exponent means repeated multiplication by the base. Whereas here we have a negative exponent which we already know that the negative sign on an exponent means the reciprocal of the given term. We can also say it as a negative exponent means repeated division by the base. Generally, to cube, a number, just multiply the number three times, as similar to squaring. For squaring a number we will multiply the number two times. Here, we have repeated multiplication of base so we can write the number on terms in exponent form i.e., $ {2^3} $ , which we have seen in this problem.