Solve for x 25^x*5^(x+3)=625^(x-7)

$25^{x} \cdot 5^{x + 3} = 625^{x - 7}$

Rewrite $25$ as $5^{2}$ .

${(5^{2})}^{x} \cdot 5^{x + 3} = 625^{x - 7}$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$5^{2 x} \cdot 5^{x + 3} = 625^{x - 7}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$5^{2 x + x + 3} = 625^{x - 7}$

Add $2 x$ and $x$ .

$5^{3 x + 3} = 625^{x - 7}$

Create equivalent expressions in the equation that all have equal bases.

$5^{3 x + 3} = 5^{4 (x - 7)}$

Since the bases are the same, then two expressions are only equal if the exponents are also equal.

$3 x + 3 = 4 (x - 7)$

Solve for $x$ .

Simplify $4 (x - 7)$ .

Rewrite.

$3 x + 3 = 0 + 0 + 4 (x - 7)$

Simplify by adding zeros.

$3 x + 3 = 4 (x - 7)$

Apply the distributive property.

$3 x + 3 = 4 x + 4 \cdot - 7$

Multiply $4$ by $- 7$ .

$3 x + 3 = 4 x - 28$

Move all terms containing $x$ to the left side of the equation.

Subtract $4 x$ from both sides of the equation.

$3 x + 3 - 4 x = - 28$

Subtract $4 x$ from $3 x$ .

$- x + 3 = - 28$

Move all terms not containing $x$ to the right side of the equation.

Subtract $3$ from both sides of the equation.

$- x = - 28 - 3$

Subtract $3$ from $- 28$ .

$- x = - 31$

Divide each term in $- x = - 31$ by $- 1$ and simplify.

Divide each term in $- x = - 31$ by $- 1$ .

$\frac{- x}{- 1} = \frac{- 31}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{- 31}{- 1}$

Divide $x$ by $1$ .

$x = \frac{- 31}{- 1}$

Simplify the right side.

Divide $- 31$ by $- 1$ .

$x = 31$