Solve the Rational Equation for x (2^(2x+2))(2^(3x-7))=2^25

$(2^{2 x + 2}) (2^{3 x - 7}) = 2^{25}$

Multiply $2^{2 x + 2}$ by $2^{3 x - 7}$ by adding the exponents.

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

$2^{2 x + 2 + 3 x - 7} = 2^{25}$

Add $2 x$ and $3 x$ .

$2^{5 x + 2 - 7} = 2^{25}$

Subtract $7$ from $2$ .

$2^{5 x - 5} = 2^{25}$

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

$5 x - 5 = 25$

Solve for $x$ .

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

Add $5$ to both sides of the equation.

$5 x = 25 + 5$

Add $25$ and $5$ .

$5 x = 30$

Divide each term in $5 x = 30$ by $5$ and simplify.

Divide each term in $5 x = 30$ by $5$ .

$\frac{5 x}{5} = \frac{30}{5}$

Simplify the left side.

Cancel the common factor of $5$ .

Cancel the common factor.

$\frac{5 x}{5} = \frac{30}{5}$

Divide $x$ by $1$ .

$x = \frac{30}{5}$

Simplify the right side.

Divide $30$ by $5$ .

$x = 6$