Question

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

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 solution:
Given, $\dfrac{{{5^m}}}{{{5^{ - 3}}}} = {5^5}$.
To evaluate this value we should follow the law of exponent.
We can simplify the expression $\dfrac{{{5^m}}}{{{5^{ - 3}}}} = {5^5}$, as follows,
$\dfrac{{{5^m}}}{{{5^{ - 3}}}} = {5^m} \times \dfrac{1}{{{5^{ - 3}}}}$
As, $\dfrac{1}{{{a^{ - 1}}}} = a$, so,
$\dfrac{1}{{{5^{ - 3}}}} = {5^3}$
The L.H.S of the expression can be more simplified as,
$\dfrac{{{5^m}}}{{{5^{ - 3}}}} = {5^m} \times \dfrac{1}{{{5^{ - 3}}}} = {5^m} \times {5^3}$
Here, we need to observe that, if there is any expression like ${a^m} \times {a^n}$, then we need add the powers of the non-zero integers, i.e., ${a^m} \times {a^n} = {a^{m + n}}$.
So, ${5^m} \times {5^3} = {\left( 5 \right)^{m + 3}}$.
Therefore, the whole expression $\dfrac{{{5^m}}}{{{5^{ - 3}}}} = {5^5}$, can be simplified as,
${\left( 5 \right)^m}^{ + 3} = {5^5}$
Now, to find the value of m, we need to compare the power coefficients of L.H.S and R.H.S, as the base coefficient is same on both the sides.
So,
$
  m + 3 = 5 \\
  m = 5 - 3 \\
  m = 2 \\
 $
Therefore, the value of m is 2.


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.