Solve for x (1/25)^(2x+3)=125^(9x+8)

${(\frac{1}{25})}^{2 x + 3} = 125^{9 x + 8}$

Step 1

Apply the product rule to $\frac{1}{25}$ .

$\frac{1^{2 x + 3}}{25^{2 x + 3}} = 125^{9 x + 8}$

Step 2

One to any power is one.

$\frac{1}{25^{2 x + 3}} = 125^{9 x + 8}$

Step 3

Move $25^{2 x + 3}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

$25^{- (2 x + 3)} = 125^{9 x + 8}$

Step 4

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

$5^{2 (- (2 x + 3))} = 5^{3 (9 x + 8)}$

Step 5

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

$2 (- (2 x + 3)) = 3 (9 x + 8)$

Step 6

Solve for $x$ .

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

Start simplifying.

$0 + 0 + 2 (- (2 x + 3)) = 3 (9 x + 8)$

Simplify by adding zeros.

$2 (- (2 x + 3)) = 3 (9 x + 8)$

Apply the distributive property.

$2 (- (2 x) - 1 \cdot 3) = 3 (9 x + 8)$

Multiply.

Multiply $2$ by $- 1$ .

$2 (- 2 x - 1 \cdot 3) = 3 (9 x + 8)$

Multiply $- 1$ by $3$ .

$2 (- 2 x - 3) = 3 (9 x + 8)$

Apply the distributive property.

$2 (- 2 x) + 2 \cdot - 3 = 3 (9 x + 8)$

Multiply.

Multiply $- 2$ by $2$ .

$- 4 x + 2 \cdot - 3 = 3 (9 x + 8)$

Multiply $2$ by $- 3$ .

$- 4 x - 6 = 3 (9 x + 8)$

Simplify $3 (9 x + 8)$ .

Apply the distributive property.

$- 4 x - 6 = 3 (9 x) + 3 \cdot 8$

Multiply.

Multiply $9$ by $3$ .

$- 4 x - 6 = 27 x + 3 \cdot 8$

Multiply $3$ by $8$ .

$- 4 x - 6 = 27 x + 24$

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

Subtract $27 x$ from both sides of the equation.

$- 4 x - 6 - 27 x = 24$

Subtract $27 x$ from $- 4 x$ .

$- 31 x - 6 = 24$

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

Add $6$ to both sides of the equation.

$- 31 x = 24 + 6$

Add $24$ and $6$ .

$- 31 x = 30$

Divide each term in $- 31 x = 30$ by $- 31$ and simplify.

Divide each term in $- 31 x = 30$ by $- 31$ .

$\frac{- 31 x}{- 31} = \frac{30}{- 31}$

Simplify the left side.

Cancel the common factor of $- 31$ .

Cancel the common factor.

$\frac{- 31 x}{- 31} = \frac{30}{- 31}$

Divide $x$ by $1$ .

$x = \frac{30}{- 31}$

Simplify the right side.

Move the negative in front of the fraction.

$x = - \frac{30}{31}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{30}{31}$

Decimal Form:

$x = - 0.96774193 \dots$

Solve for x (1/25)^(2x+3)=125^(9x+8)

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