To solve the equation [tex]\( 125^{y + 1} + 5^{3y} = 630 \)[/tex] for [tex]\( y \)[/tex], follow these steps:
1. Express the terms with the same base: Notice that 125 and 5 are powers of the same base:
[tex]\[ 125 = 5^3 \][/tex]
So, rewrite the equation using [tex]\( 5 \)[/tex]:
[tex]\[ (5^3)^{y + 1} + 5^{3y} = 630 \][/tex]
2. Simplify the exponents:
[tex]\[ (5^3)^{y + 1} = 5^{3(y + 1)} = 5^{3y + 3} \][/tex]
Thus, the equation becomes:
[tex]\[ 5^{3y + 3} + 5^{3y} = 630 \][/tex]
3. Factor out the common term: The common term is [tex]\( 5^{3y} \)[/tex]. Factor this out:
[tex]\[ 5^{3y} \cdot 5^3 + 5^{3y} = 630 \][/tex]
[tex]\[ 5^{3y} \cdot 125 + 5^{3y} = 630 \][/tex]
4. Combine like terms:
[tex]\[ 5^{3y} (125 + 1) = 630 \][/tex]
[tex]\[ 5^{3y} \cdot 126 = 630 \][/tex]
5. Solve for the exponential term:
[tex]\[ 5^{3y} = \frac{630}{126} \][/tex]
[tex]\[ 5^{3y} = 5 \][/tex]
6. Simplify the right-hand side:
[tex]\[ 5^{3y} = 5^1 \][/tex]
7. Equate the exponents: Since the bases are the same, the exponents must be equal:
[tex]\[ 3y = 1 \][/tex]
8. Solve for [tex]\( y \)[/tex]:
[tex]\[ y = \frac{1}{3} \][/tex]
Therefore, the solution to the equation [tex]\( 125^{y+1} + 5^{3y} = 630 \)[/tex] is:
[tex]\[ y = \frac{1}{3} \][/tex]
