Solve for x 25^(10x+8)=(1/125)^(4-2x)

$25^{10 x + 8} = {(\frac{1}{125})}^{4 - 2 x}$

Step 1

Apply the product rule to $\frac{1}{125}$ .

$25^{10 x + 8} = \frac{1^{4 - 2 x}}{125^{4 - 2 x}}$

Step 2

One to any power is one.

$25^{10 x + 8} = \frac{1}{125^{4 - 2 x}}$

Step 3

Move $125^{4 - 2 x}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

$25^{10 x + 8} = 125^{- (4 - 2 x)}$

Step 4

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

$5^{2 (10 x + 8)} = 5^{3 (- (4 - 2 x))}$

Step 5

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

$2 (10 x + 8) = 3 (- (4 - 2 x))$

Step 6

Solve for $x$ .

Simplify $2 (10 x + 8)$ .

Rewrite.

$0 + 0 + 2 (10 x + 8) = 3 (- (4 - 2 x))$

Simplify by adding zeros.

$2 (10 x + 8) = 3 (- (4 - 2 x))$

Apply the distributive property.

$2 (10 x) + 2 \cdot 8 = 3 (- (4 - 2 x))$

Multiply.

Multiply $10$ by $2$ .

$20 x + 2 \cdot 8 = 3 (- (4 - 2 x))$

Multiply $2$ by $8$ .

$20 x + 16 = 3 (- (4 - 2 x))$

Simplify $3 (- (4 - 2 x))$ .

Apply the distributive property.

$20 x + 16 = 3 (- 1 \cdot 4 - (- 2 x))$

Multiply.

Multiply $- 1$ by $4$ .

$20 x + 16 = 3 (- 4 - (- 2 x))$

Multiply $- 2$ by $- 1$ .

$20 x + 16 = 3 (- 4 + 2 x)$

Apply the distributive property.

$20 x + 16 = 3 \cdot - 4 + 3 (2 x)$

Multiply.

Multiply $3$ by $- 4$ .

$20 x + 16 = - 12 + 3 (2 x)$

Multiply $2$ by $3$ .

$20 x + 16 = - 12 + 6 x$

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

Subtract $6 x$ from both sides of the equation.

$20 x + 16 - 6 x = - 12$

Subtract $6 x$ from $20 x$ .

$14 x + 16 = - 12$

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

Subtract $16$ from both sides of the equation.

$14 x = - 12 - 16$

Subtract $16$ from $- 12$ .

$14 x = - 28$

Divide each term in $14 x = - 28$ by $14$ and simplify.

Divide each term in $14 x = - 28$ by $14$ .

$\frac{14 x}{14} = \frac{- 28}{14}$

Simplify the left side.

Cancel the common factor of $14$ .

Cancel the common factor.

$\frac{14 x}{14} = \frac{- 28}{14}$

Divide $x$ by $1$ .

$x = \frac{- 28}{14}$

Simplify the right side.

Divide $- 28$ by $14$ .

$x = - 2$

Solve for x 25^(10x+8)=(1/125)^(4-2x)

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