Question

How do you solve ${{5}^{x+1}}=125$?

Answer 1

This type of problem can be solved by converting the base to a common term and then equating the powers with the variable. First, we have to consider the right-hand side of the given equation. Write 125 as the cube of 5. Now the bases to the LHS and RHS are the same. Using the rule ‘if${{a}^{n}}={{a}^{m}}$, then n=m’, we get x+1=3. Subtract 1 from both the sides of the expression and do necessary calculations to get the value of x which is the required answer.

Complete step-by-step solution:
According to the question, we are asked to solve ${{5}^{x+1}}=125$.
We have been given the equation is ${{5}^{x+1}}=125$. ---------(1)
Let us first consider the right-hand side of the given equation (1).
RHS=125
We know that $125=5\times 5\times 5$.
Therefore, we can write 125 as the cube of 5 that is ${{5}^{3}}$.
Therefore, RHS=${{5}^{3}}$.
Substitute the RHS in equation (1).
$\Rightarrow {{5}^{x+1}}={{5}^{3}}$
We find that the LHS and RHS have the same term 5 with the power difference.
We know that, if ${{a}^{n}}={{a}^{m}}$, then n=m.
Using this rule in the above expression, we get
$x+1=3$
Subtract 1 from both the sides of the expression, we get
$x+1-1=3-1$
We know that terms with the same magnitude and opposite signs cancel out.
Therefore, we get
$x=3-1$
On further simplification, we get
$x=2$
Hence, the value of x in ${{5}^{x+1}}=125$ is 2.

Note: Whenever we get such types of problems, we should always compare the right-hand side of the equation with the left-hand side of the equation. Convert the LHS and RHS of the equation in such a way that we can equate their powers. Avoid calculation mistakes based on sign conventions. Do not neglect the negative sign in the power.