Question

Simplify $ {\left( {\dfrac{5}{8}} \right)^{ - 7}} \times {\left( {\dfrac{8}{5}} \right)^{ - 5}} $ ?

Answer 1

Hint: To simplify this question , we need to solve it step by step . . We must know about the exponents . The exponent of a number says how many times to use the number in a multiplication. Exponents can also be called as Power. For example = $ {5^2} $ could be called “ $ 5 $ to the power “ $ 2 $ ” that means the “ $ 2 $ ” says to use $ 5 $ twice in a multiplication. So, 5 is the base and 2 is the power . For an instance , $ {5^2} = 5 \times 5 = 25 $ We should know that the exponent having -1 as power is considered to be $ $ $ \dfrac{1}{x} $ . So , by rewriting all the powers having negative exponents can be rewritten by using only positive exponents , we will rewrite it similarly and try to simplify by cancelling out the common factors to get the required result .

Complete step-by-step answer:
In order to solve the expression , write it the way it is given and then arrange according to the positive exponents by making them the denominator .
As per our given Question ,
 $ {\left( {\dfrac{5}{8}} \right)^{ - 7}} \times {\left( {\dfrac{8}{5}} \right)^{ - 5}} $
The exponents having $ {x^{ - 1}} $ , -1 as power is considered to be $ $ $ \dfrac{1}{x} $ . Convert only one exponent to positive of the two fractions given , so that simplification would be easier .
Rewrite the above accordingly as given in the expression , we have =
\[
  {\left( {\dfrac{5}{8}} \right)^{ - 1 \times 7}} \times {\left( {\dfrac{8}{5}} \right)^{ - 1 \times 5}} \\
   \Rightarrow {\left( {\dfrac{8}{5}} \right)^7} \times {\left( {\dfrac{8}{5}} \right)^{ - 1 \times 5}} \;
 \]
Now, after this we will apply one property of exponents i.e. Product rule which states that whenever the bases will be same then the powers or the exponents gets added .
 \[{a^{\;n}}\; \cdot \;{a^{\;m}}\; = \;{a^{\;n + m}}\]
\[
   \Rightarrow {\left( {\dfrac{8}{5}} \right)^7} \times {\left( {\dfrac{8}{5}} \right)^{ - 1 \times 5}} \\
   \Rightarrow {\left( {\dfrac{8}{5}} \right)^7} \times {\left( {\dfrac{8}{5}} \right)^{ - 5}} \\
   \Rightarrow {\left( {\dfrac{8}{5}} \right)^{7 + ( - 5)}} \\
   \Rightarrow {\left( {\dfrac{8}{5}} \right)^{7 - 5}} \\
   \Rightarrow {\left( {\dfrac{8}{5}} \right)^2} \;
 \]
Now further simplifying the expression \[{\left( {\dfrac{8}{5}} \right)^2}\]we will get-
\[{\left( {\dfrac{8}{5}} \right)^2} = \dfrac{{8 \times 8}}{{5 \times 5}} = \dfrac{{64}}{{25}}\]
Therefore , the required solution is \[\dfrac{{64}}{{25}}\].
So, the correct answer is “s \[\dfrac{{64}}{{25}}\]”.

Note: Always remember you can perform calculations only between like terms .
Always try to work on the positive exponents .
The expression is having hidden multiplication when written altogether .
Do not forget to verify the exponents solved correctly .
Always try to cancel out the similar terms for the solution of simplification .