Question

Find the value of x in the given expression, ${{3}^{2x-8}}\times {{5}^{x-3}}=225$.

Answer 1

Hint: We will first factorize 225 and write it in terms of the power of 3 and 5. We will then apply the concept that equal bases have equal exponents and proceed to find the answer.

Complete step-by-step solution
In the question, we are given an expression, ${{3}^{2x-8}}\times {{5}^{x-3}}=225$ and have been asked to find the value of x. We will start solving this by first factoring 225. So, we will factorise 225 as follows,
$\begin{align}
  & 3\left| \!{\underline {\,
  225 \,}} \right. \\
 & 3\left| \!{\underline {\,
  75 \,}} \right. \\
 & 5\left| \!{\underline {\,
  25 \,}} \right. \\
 & \text{ }5 \\
\end{align}$
So, as we can see, we can write 22 5 as, $3\times 3\times 5\times 5$. We know that this can also be written in the form of powers as, ${{3}^{2}}\times {{5}^{2}}$. The equation given to us was, ${{3}^{2x-8}}\times {{5}^{x-3}}=225$, so we can rewrite 225 in the form of power that we have obtained. So, we can write the equation as below,
${{3}^{2x-8}}\times {{5}^{x-3}}={{3}^{2}}\times {{5}^{2}}$
Now, here, we will compare the powers of the similar bases only, so we will have two equations, one for base 3 and one for base 5. Let us take the first equation of the base 3, so we have,
${{3}^{2x-8}}={{3}^{2}}$
We know that the rule of exponents states that, ‘equal bases have equal exponents’. So applying that rule here, we will get the equation as,
$2x-8=2$
Adding 8 to both the sides, we will get,
$\begin{align}
  & 2x-8+8=2+8 \\
 & \Rightarrow 2x=10 \\
 & \Rightarrow x=\dfrac{10}{2} \\
 & \Rightarrow x=5 \\
\end{align}$
So, we get the value of x as 5. Now let us consider the second equation for base 5. So, we have the equation as,
${{5}^{x-3}}={{5}^{2}}$
Again, we will apply the rule of exponents here. So, we get the equation as,
$x-3=2$
Adding 3 on both sides of the equation, we get,
$\begin{align}
  & x-3+3=2+3 \\
 & \Rightarrow x=5 \\
\end{align}$
So, we get the value of x as 5.
Thus, the value of x in the given expression, ${{3}^{2x-8}}\times {{5}^{x-3}}=225$ is 5.

Note: The students should know that if they get two different values for x while solving both the equations of the bases, then their answer would be incorrect and they should go through the steps again. They can also check their answer by substituting the value of x in the given expression to see whether both the sides of the given equation are the same or not.