Question

By what number should \[{( - 8)^{ - 3}}\] be multiplied so that the product may be equal to \[{( - 6)^{ - 3}}\]?

Answer 1

Hint: As we are asked to find out the number in the given equation we have to take an arbitrary no, x to solve the problem. We also have to use the exponential formulas to solve the process further.
The exponential formula is: \[{a^{ - m}} = \dfrac{1}{{{a^m}}}\]

Complete step-by-step solution:
First, we have to take an arbitrary number to be X to solve the problem
According to the question, we are asked what multiplies X to get the given result.
Putting them into the equation we get:
\[X \times {( - 8)^{ - 3}} = {( - 6)^{ - 3}}\]
Solving the equation for x we get:
\[X = \dfrac{{{{( - 6)}^{ - 3}}}}{{{{( - 8)}^{ - 3}}}}\]
Now by the exponential formula, we know that: \[{a^{ - m}} = \dfrac{1}{{{a^m}}}\] so we have
\[
  {( - 6)^{ - 3}} = \dfrac{1}{{{{( - 6)}^3}}} \\
  {( - 8)^{ - 3}} = \dfrac{1}{{{{( - 8)}^3}}} \\
 \]
\[
   = \dfrac{1}{{{{( - 6)}^3}}} \times \dfrac{{{{( - 8)}^3}}}{1} \\
   = \dfrac{{{{( - 8)}^3}}}{{{{( - 6)}^3}}} \\
 \]
Hence X \[ = \dfrac{{{{( - 8)}^3}}}{{{{( - 6)}^3}}}\]
Solving it further we have:
 \[{( - 8)^3} = ( - 8) \times ( - 8) \times ( - 8) = - 512\]
\[{( - 6)^3} = ( - 6) \times ( - 6) \times ( - 6) = - 216\]
Substituting in X we get
\[
  X = \dfrac{{ - 512}}{{ - 216}} \\
 \Rightarrow X = \dfrac{{64}}{{27}} \\
 \]
Here negative gets canceled with negative and gives a positive number.
Hence \[X = \dfrac{{64}}{{27}}\] is the number when multiplied by \[{( - 8)^{ - 3}}\] gives a product which is equal to \[{( - 6)^{ - 3}}\].
Additional Information: Whenever a question asks you to find out a number it is always recommended to take an arbitrary number X to solve the question. Substituting X according to what is asked in the question can help us solve the question further. Knowledge of exponents and the use of their formulas are equally important. If one is accustomed to the exponential formulas he/she can easily simplify the question and reach the final answer.

Note: Looking at the question we can see that it is in exponential form, hence exponential formulas will be used. \[{a^{ - m}} = \dfrac{1}{{{a^m}}}\] Is an essential formula of all times and helps in dealing with problems having negative signs.