Question

Find the value of $m$ : ${\left( {\dfrac{2}{9}} \right)^3} \times {\left( {\dfrac{2}{9}} \right)^6} = {\left( {\dfrac{2}{9}} \right)^{2m - 1}}$

Answer 1

Hint: In this question we can see that the above expression is of the exponential form. We can also see that all the bases of the left hand side and right hand side of the equation are the same. So we will apply the basic exponential formulas to solve this.
We will use the formula:
${(a)^m} \times {(a)^n} = {(a)^{m + n}}$.
This is the product rule of exponent. It says that in multiplication if the two powers have the same base, then we can add the exponents.

Complete step by step answer:
Let us understand the definition of exponential form.
We know that a mathematical expression that has a number raised to the power of another number, then it is called an exponential expression. If we have ${(5)^2}$ , then $5$ is the base and $2$ is the exponential power.
Now in the question we have
${\left( {\dfrac{2}{9}} \right)^3} \times {\left( {\dfrac{2}{9}} \right)^6} = {\left( {\dfrac{2}{9}} \right)^{2m - 1}}$
In the left hand side of the equation, we will apply the product rule of exponent. Since we have the same base $\dfrac{2}{9}$ .
By applying the formula we can write
 ${\left( {\dfrac{2}{9}} \right)^{3 + 6}} = {\left( {\dfrac{2}{9}} \right)^{2m - 1}}$
We will add the powers, so we have
${\left( {\dfrac{2}{9}} \right)^9} = {\left( {\dfrac{2}{9}} \right)^{2m - 1}}$
Now we know that if the bases on both sides of the equation are the same, we can equate the powers.
Therefore we can write the expression as
$9 = 2m - 1$
We will now simplify this:
$2m = 9 + 1 \Rightarrow 2m = 10$
It gives us value
$m = \dfrac{{10}}{2} = 5$
Hence the required value of $m$ is $5$ .

Note:
 Before solving these kinds of questions we should always remember the exponential formulas. We should know that if we have to divide the expression that has the same base, then the power gets subtracted.
It can be written as
 $\dfrac{{{a^m}}}{{{a^n}}} = {a^{m - n}}$ , here we can see that the base of both the terms is same.
 Any expression with exponential power zero is always equal to $1$, i.e.
${a^0} = 1$ .