Question

Evaluate
I.$\dfrac{{{8^{ - 1}} \times {5^3}}}{{{2^{ - 4}}}}$
II.$\left( {{5^{ - 1}} \times {2^{ - 1}}} \right) \times {6^{ - 1}}$

Answer 1

Hint- As, it is a problem of exponents and power, we need to use the concept of ${a^{ - 1}} = \dfrac{1}{a}$, $\dfrac{1}{{{b^{ - 1}}}} = b$, so when the expression is fragmented in smaller parts these operation can be done. 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, $\dfrac{{{8^{ - 1}} \times {5^3}}}{{{2^{ - 4}}}}$.
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 $\dfrac{1}{{{b^{ - 1}}}} = b$.
Expression $\dfrac{{{8^{ - 1}} \times {5^3}}}{{{2^{ - 4}}}}$ can be more simplified as follows:
$\dfrac{{{8^{ - 1}} \times {5^3}}}{{{2^{ - 4}}}} = {8^{ - 1}} \times {5^3} \times \dfrac{1}{{{2^{ - 4}}}}$
As, ${8^{ - 1}} = \dfrac{1}{8} = \dfrac{1}{{{2^3}}}$ and $\dfrac{1}{{{2^{ - 4}}}} = {2^4}$
So,
$\dfrac{{{8^{ - 1}} \times {5^3}}}{{{2^{ - 4}}}} = \dfrac{1}{{{2^3}}} \times {5^3} \times {2^4}$
Here, we need to observe that ${2^3}$ will be cancelled by ${2^4}$, hence the remainder will be 2.
Therefore,
$
  \dfrac{{{8^{ - 1}} \times {5^3}}}{{{2^{ - 4}}}} = {5^3} \times 2 \\
   = 5 \times 5 \times 5 \times 2 \\
   = 250 \\
 $
So, on evaluating the value of expression $\dfrac{{{8^{ - 1}} \times {5^3}}}{{{2^{ - 4}}}}$ its value comes out to be 250.
Given, $\left( {{5^{ - 1}} \times {2^{ - 1}}} \right) \times {6^{ - 1}}$.
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 $\dfrac{1}{{{b^{ - 1}}}} = b$.
Expression $\left( {{5^{ - 1}} \times {2^{ - 1}}} \right) \times {6^{ - 1}}$ can be more simplified as follows:
$\left( {{5^{ - 1}} \times {2^{ - 1}}} \right) \times {6^{ - 1}} = \left( {\dfrac{1}{5} \times \dfrac{1}{2}} \right) \times \dfrac{1}{6}$
Now, we have to multiply the terms with each other.
$
  \left( {\dfrac{1}{5} \times \dfrac{1}{2}} \right) \times \dfrac{1}{6} = \left( {\dfrac{1}{{5 \times 2}}} \right) \times \dfrac{1}{6} \\
   = \left( {\dfrac{1}{{10}}} \right) \times \dfrac{1}{6} \\
   = \dfrac{1}{{10 \times 6}} \\
   = \dfrac{1}{{60}} \\
 $
So, on evaluating the value of expression $\left( {{5^{ - 1}} \times {2^{ - 1}}} \right) \times {6^{ - 1}}$ its value comes out to be $\dfrac{1}{{60}}$.


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.