Let's solve the equation [tex]\( 5^x + 5^{x+1} = 750 \)[/tex] step-by-step.
1. Rewrite the equation involving exponents:
The term [tex]\( 5^{x+1} \)[/tex] can be rewritten as [tex]\( 5 \cdot 5^x \)[/tex]. Let's substitute this back into the equation:
[tex]\[
5^x + 5 \cdot 5^x = 750
\][/tex]
2. Factor out the common term:
Notice that [tex]\( 5^x \)[/tex] is a common factor in the left-hand side of the equation. Factor [tex]\( 5^x \)[/tex] out:
[tex]\[
5^x (1 + 5) = 750
\][/tex]
3. Simplify inside the parentheses:
Calculate the sum inside the parentheses:
[tex]\[
5^x \cdot 6 = 750
\][/tex]
4. Isolate [tex]\( 5^x \)[/tex]:
Divide both sides of the equation by 6 to solve for [tex]\( 5^x \)[/tex]:
[tex]\[
5^x = \frac{750}{6}
\][/tex]
5. Simplify the division:
Perform the division:
[tex]\[
5^x = 125
\][/tex]
6. Equate to the base:
Recognize that 125 can be written as a power of 5:
[tex]\[
125 = 5^3
\][/tex]
So, we can write:
[tex]\[
5^x = 5^3
\][/tex]
7. Set the exponents equal:
Since the bases are the same (both are 5), we can set the exponents equal to each other:
[tex]\[
x = 3
\][/tex]
Therefore, the value of [tex]\( x \)[/tex] is [tex]\(\boxed{3}\)[/tex].
