Question

Which of the following can also be represented as ${{\left( -5 \right)}^{-1}}$
(a) ${{\left( -5 \right)}^{3}}\times {{\left( -5 \right)}^{2}}$
(b) ${{\left( -5 \right)}^{-3}}\times {{\left( -5 \right)}^{2}}$
(c) ${{\left( -5 \right)}^{3}}\times {{\left( -5 \right)}^{-2}}$
(d) ${{\left( -5 \right)}^{-3}}\times {{\left( -5 \right)}^{-2}}$

Answer 1

Hint: To solve this question, take all the options one by one and simplify them by using the formula ${{x}^{a}}\times {{x}^{b}}={{x}^{a+b}}$. For each option, check whether its simplification is equal to ${{\left( -5 \right)}^{-1}}$ or not. If yes, then that will be the final answer.

Complete step-by-step solution -

In this question, we need to find which expression among the given options can also be represented as ${{\left( -5 \right)}^{-1}}$.
In this question, we will use the property that if a number x raised to the power of a is multiplied with the same number x raised to the power of b, then the result is x raised to the power of the sum of a and b.
i.e. ${{x}^{a}}\times {{x}^{b}}={{x}^{a+b}}$
Using this property, we will reduce all the options and check whether they are equal to ${{\left( -5 \right)}^{-1}}$ or not.
For first option: ${{\left( -5 \right)}^{3}}\times {{\left( -5 \right)}^{2}}$
Using the property above, we will get the following:
${{\left( -5 \right)}^{3}}\times {{\left( -5 \right)}^{2}}={{\left( -5 \right)}^{3+2}}={{\left( -5 \right)}^{5}}$
This is not equal to ${{\left( -5 \right)}^{-1}}$. So, this is not the answer.
For second option: ${{\left( -5 \right)}^{-3}}\times {{\left( -5 \right)}^{2}}$
Using the property above, we will get the following:
${{\left( -5 \right)}^{-3}}\times {{\left( -5 \right)}^{2}}={{\left( -5 \right)}^{-3+2}}={{\left( -5 \right)}^{-1}}$
This is equal to ${{\left( -5 \right)}^{-1}}$. So, this is the answer.
For third option: ${{\left( -5 \right)}^{3}}\times {{\left( -5 \right)}^{-2}}$
Using the property above, we will get the following:
${{\left( -5 \right)}^{3}}\times {{\left( -5 \right)}^{-2}}={{\left( -5 \right)}^{3-2}}={{\left( -5 \right)}^{1}}$
This is not equal to ${{\left( -5 \right)}^{-1}}$. So, this is not the answer.
For fourth option: ${{\left( -5 \right)}^{-3}}\times {{\left( -5 \right)}^{-2}}$
Using the property above, we will get the following:
${{\left( -5 \right)}^{-3}}\times {{\left( -5 \right)}^{-2}}={{\left( -5 \right)}^{-3-2}}={{\left( -5 \right)}^{-5}}$
This is not equal to ${{\left( -5 \right)}^{-1}}$. So, this is not the answer.
Hence, only option (b) is correct.

Note: In this question, it is very important to know the following property: that if a number x raised to the power of a is multiplied with the same number x raised to the power of b, then the result is x raised to the power of the sum of a and b. i.e. ${{x}^{a}}\times {{x}^{b}}={{x}^{a+b}}$.