Question

Evaluate
I. $ {\left( {\dfrac{1}{3}} \right)^{ - 1}} - {\left( {\dfrac{1}{4}} \right)^{ - 1}} $
II. $ {\left( {\dfrac{5}{8}} \right)^{ - 7}} \times {\left( {\dfrac{8}{5}} \right)^{ - 4}} $

Answer 1

Hint: As, it is a problem of exponents and power, we need to use the concept of $ {a^{ - 1}} = \dfrac{1}{a} $ , $ {\left( {\dfrac{1}{b}} \right)^{ - 1}} = b $ . Whenever any base has power of negative number, then the reciprocal of the base is taken, as by taking the reciprocal of any number then its power sign is changed.

Complete step-by-step answer:
Given, $ {\left( {\dfrac{1}{3}} \right)^{ - 1}} - {\left( {\dfrac{1}{4}} \right)^{ - 1}} $ .
To evaluate this value we should follow the law of exponent.
We need to find the value of the above expression.
To evaluate this value we should follow the law of exponent.
As, $ {a^{ - 1}} = \dfrac{1}{a} $ and $ {\left( {\dfrac{1}{b}} \right)^{ - 1}} = b $ .
Therefore, $ {\left( {\dfrac{1}{3}} \right)^{ - 1}} - {\left( {\dfrac{1}{4}} \right)^{ - 1}} $ can be simplified as,
 $
\Rightarrow {\left( {\dfrac{1}{3}} \right)^{ - 1}} - {\left( {\dfrac{1}{4}} \right)^{ - 1}} = 3 - 4 \\
\Rightarrow {\left( {\dfrac{1}{3}} \right)^{ - 1}} - {\left( {\dfrac{1}{4}} \right)^{ - 1}} = - 1 \\
  $
So, the value of expression $ {\left( {\dfrac{1}{3}} \right)^{ - 1}} - {\left( {\dfrac{1}{4}} \right)^{ - 1}} $ is $ - 1 $
Given, $ {\left( {\dfrac{5}{8}} \right)^{ - 7}} \times {\left( {\dfrac{8}{5}} \right)^{ - 4}} $ .
To evaluate this value we should follow the law of exponent.
We need to find the value of the above expression.
To evaluate this value we should follow the law of exponent.
As, $ {a^{ - 1}} = \dfrac{1}{a} $ and $ {\left( {\dfrac{1}{b}} \right)^{ - 1}} = b $ .
Same thing happens with the expression $ {\left( {\dfrac{a}{b}} \right)^{ - 1}} = \dfrac{b}{a} $ .
Therefore, $ {\left( {\dfrac{5}{8}} \right)^{ - 7}} \times {\left( {\dfrac{8}{5}} \right)^{ - 4}} $ can be simplified as,
 $
\Rightarrow {\left( {\dfrac{5}{8}} \right)^{ - 7}} \times {\left( {\dfrac{8}{5}} \right)^{ - 4}} = {\left( {\dfrac{8}{5}} \right)^7} \times {\left( {\dfrac{5}{8}} \right)^4} $
Expression, $ {\left( {\dfrac{8}{5}} \right)^7} \times {\left( {\dfrac{5}{8}} \right)^4} $ can be written as $ \dfrac{{{8^7}}}{{{5^7}}} \times \dfrac{{{5^4}}}{{{8^4}}} $ .
Here, we need to notice that $ {8^4} $ will be cancelled out by $ {8^7} $ and the remainder will be $ {8^3} $ and same thing will happen with $ {5^4} $ , it will be cancelled by $ {5^7} $ and the remainder will be $ {5^3} $
So, the simplified expression will be,
 $
\Rightarrow {\left( {\dfrac{5}{8}} \right)^{ - 7}} \times {\left( {\dfrac{8}{5}} \right)^{ - 4}} = {\left( {\dfrac{8}{5}} \right)^7} \times {\left( {\dfrac{5}{8}} \right)^4} = \dfrac{{{8^7}}}{{{5^7}}} \times \dfrac{{{5^4}}}{{{8^4}}} = \dfrac{{{8^3}}}{{{5^3}}} = \dfrac{{8 \times 8 \times 8}}{{5 \times 5 \times 5}} = \dfrac{{512}}{{125}} $
Therefore, the value of the expression $ {\left( {\dfrac{5}{8}} \right)^{ - 7}} \times {\left( {\dfrac{8}{5}} \right)^{ - 4}} $ will be $ \dfrac{{512}}{{125}} $ .

Note: This question is of the concept power and exponent. Power or exponent of any number is the number of times any particular number is being multiplied by it. Whenever in any expression there is involvement of two things base and power that number is called an exponential number.