Question
Find the value of x to make this exponential equation true. 5 to the power of x divided by 5 to the power of 5 equals 5 to the power of 12
Answers
GPT-4o mini
To solve the equation
\[
\frac{5^x}{5^5} = 5^{12},
\]
we can apply the properties of exponents. Recall that when we divide two powers with the same base, we subtract the exponents:
\[
\frac{5^a}{5^b} = 5^{a-b}.
\]
Using this property, we rewrite the left side of the equation:
\[
5^{x - 5} = 5^{12}.
\]
Since the bases are the same, we can set the exponents equal to each other:
\[
x - 5 = 12.
\]
Now, to solve for \( x \), we add 5 to both sides:
\[
x = 12 + 5,
\]
which simplifies to
\[
x = 17.
\]
Thus, the value of \( x \) that makes the equation true is
\[
\boxed{17}.
\]
\[
\frac{5^x}{5^5} = 5^{12},
\]
we can apply the properties of exponents. Recall that when we divide two powers with the same base, we subtract the exponents:
\[
\frac{5^a}{5^b} = 5^{a-b}.
\]
Using this property, we rewrite the left side of the equation:
\[
5^{x - 5} = 5^{12}.
\]
Since the bases are the same, we can set the exponents equal to each other:
\[
x - 5 = 12.
\]
Now, to solve for \( x \), we add 5 to both sides:
\[
x = 12 + 5,
\]
which simplifies to
\[
x = 17.
\]
Thus, the value of \( x \) that makes the equation true is
\[
\boxed{17}.
\]