Question

If we have $800=8\times {{10}^{8}}\times {{x}^{-\dfrac{3}{2}}}$, then the value of x is
(A). ${{10}^{2}}$
(B). ${{10}^{3}}$
(C). ${{10}^{4}}$
(D). ${{10}^{5}}$

Answer 1

Hint: In this given question, first of all we can convert 800 into the form of $8\times {{10}^{2}}$, then we can simplify it and get the value of x. Here, we need to know that ${{a}^{\dfrac{m}{n}}}={{\left( {{a}^{\dfrac{1}{n}}} \right)}^{m}}={{\left( {{a}^{m}} \right)}^{\dfrac{1}{n}}}$ and use this in order to solve this question.

Complete step-by-step solution -
In this given question, we are asked to find out the value of x from the equation $800=8\times {{10}^{8}}\times {{x}^{-\dfrac{3}{2}}}$.
In order to solve this question, we are going to use the following concept:
${{a}^{\dfrac{m}{n}}}={{\left( {{a}^{\dfrac{1}{n}}} \right)}^{m}}={{\left( {{a}^{m}} \right)}^{\dfrac{1}{n}}}............(1.1)$
The process of solving is as follows:
Writing 800 as $8\times {{10}^{2}}$, we can rewrite the equation given in the question as,
\[\begin{align}
  & 800=8\times {{10}^{8}}\times {{x}^{-\dfrac{3}{2}}} \\
 & \Rightarrow 8\times {{10}^{2}}=8\times {{10}^{8}}\times {{x}^{-\dfrac{3}{2}}} \\
 & \Rightarrow {{10}^{2}}={{10}^{8}}\times \dfrac{1}{{{x}^{\dfrac{3}{2}}}} \\
 & \Rightarrow {{x}^{\dfrac{3}{2}}}={{10}^{6}}..............(1.2) \\
\end{align}\]
Now, we know that ${{x}^{\dfrac{3}{2}}}={{\left( {{x}^{3}} \right)}^{\dfrac{1}{2}}}$ and ${{10}^{6}}={{\left( {{10}^{12}} \right)}^{\dfrac{1}{2}}}$. So, putting these values in equation 1.2, we get,
$\begin{align}
  & {{x}^{\dfrac{3}{2}}}={{10}^{6}} \\
 & \Rightarrow {{\left( {{x}^{3}} \right)}^{\dfrac{1}{2}}}={{\left( {{10}^{12}} \right)}^{\dfrac{1}{2}}} \\
 & \Rightarrow {{x}^{3}}={{10}^{12}} \\
 & \Rightarrow {{\left( {{x}^{3}} \right)}^{\dfrac{1}{3}}}={{\left( {{10}^{12}} \right)}^{\dfrac{1}{3}}}={{10}^{4}} \\
 & \Rightarrow x={{10}^{4}}..............(1.3) \\
\end{align}$
Hence, we have obtained the required answer to this given question as $x={{10}^{4}}$ from equation 1.3.
Therefore, the correct option as the solution to this question is an option (c) as it corresponds to our obtained answer ${{10}^{4}}$.

Note: We should note that in order to reduce the power of a number by n, we should take the power of both sides equal to $\dfrac{1}{n}$ so that the exponents cancel out. We should not try to take the power of the negative of the exponent on both sides as by this the exponents will not cancel out and thus we cannot find the value of the base quantity.