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Find the value of m for which ${{5}^{m}}\div {{5}^{-3}}={{5}^{5}}$.

Last updated date: 13th Jul 2024
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Hint: In order to solve this problem, we need to know the rules of the indices. The rule says that we can write the division of the two numbers in the form of the ratio. Also, we can combine the powers with a common base by adding the power and getting a change when changing from numerator to denominator and vice versa.

Complete step-by-step solution:
We have the expression ${{5}^{m}}\div {{5}^{-3}}={{5}^{5}}$ and we need to find the value of $m$.
We can write the division of the two numbers in the form of the ratio.
For example, we can write the value of the $10\div 2$ are $\dfrac{10}{2}$, where 10 is the numerator and the 2 is the denominator.
In the expression that we are given similar functions separated by the division.
Therefore, we can write in the form of numerator and denominator as follows,
  & {{5}^{m}}\div {{5}^{-3}}={{5}^{5}} \\
 & \Rightarrow \dfrac{{{5}^{m}}}{{{5}^{-3}}}={{5}^{5}} \\
Now, as the base of both numerator and the denominator are same, we can write them together by combining their power.
For example, we can write $\dfrac{{{10}^{5}}}{{{10}^{3}}}={{10}^{5-3}}={{10}^{2}}$
We need to add the power by writing the common base but while shifting from numerator to denominator the sign of the power changes and the same is the case when shifting from denominator to numerator.
Similarly, in the given expression we have two function with same base, combining the we get,
  & \dfrac{{{5}^{m}}}{{{5}^{-3}}}={{5}^{5}} \\
 & {{5}^{m+3}}={{5}^{5}} \\
Now, we have the equation where we have the same base either side.
There is only one possible way that we have the same base, it’s when the power they are raised to is also the same.
Therefore, we can write the equation as,
$m + 3 = 5$,
$m = 5 – 3 = 2$.
Therefore, the value of $m\; \text{is}\; 2$.

Note: In this problem, we need to remember to change the sign of -3 to +3 while moving from denominator to numerator. We can also solve this by taking the log on both sides. Taking log on both sides we get, $\log {{5}^{m+3}}=\log {{5}^{5}}$. We have the property that $\log {{a}^{b}}=b\log a$. By using that we get,
  & \left( m+3 \right)\log 5=5\log 5 \\
 & m+3=5 \\
Hence, we arrive at the same expression. Therefore, the value of $m\; \text{is}\; 2$.