Solve for x (5^(6x+5))/(125^(3x-1))=5^(-5x+2)

$\frac{5^{6 x + 5}}{125^{3 x - 1}} = 5^{- 5 x + 2}$

Step 1

Move $125^{3 x - 1}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

$5^{6 x + 5} \cdot 125^{- (3 x - 1)} = 5^{- 5 x + 2}$

Step 2

Rewrite $125$ as $5^{3}$ .

$5^{6 x + 5} \cdot {(5^{3})}^{- (3 x - 1)} = 5^{- 5 x + 2}$

Step 3

Multiply the exponents in ${(5^{3})}^{- (3 x - 1)}$ .

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

$5^{6 x + 5} \cdot 5^{3 (- (3 x - 1))} = 5^{- 5 x + 2}$

Apply the distributive property.

$5^{6 x + 5} \cdot 5^{3 (- (3 x) - - 1)} = 5^{- 5 x + 2}$

Multiply $3$ by $- 1$ .

$5^{6 x + 5} \cdot 5^{3 (- 3 x - - 1)} = 5^{- 5 x + 2}$

Multiply $- 1$ by $- 1$ .

$5^{6 x + 5} \cdot 5^{3 (- 3 x + 1)} = 5^{- 5 x + 2}$

Apply the distributive property.

$5^{6 x + 5} \cdot 5^{3 (- 3 x) + 3 \cdot 1} = 5^{- 5 x + 2}$

Multiply $- 3$ by $3$ .

$5^{6 x + 5} \cdot 5^{- 9 x + 3 \cdot 1} = 5^{- 5 x + 2}$

Multiply $3$ by $1$ .

$5^{6 x + 5} \cdot 5^{- 9 x + 3} = 5^{- 5 x + 2}$

Step 4

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

$5^{6 x + 5 - 9 x + 3} = 5^{- 5 x + 2}$

Step 5

Subtract $9 x$ from $6 x$ .

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

Step 6

Add $5$ and $3$ .

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

Step 7

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

$- 3 x + 8 = - 5 x + 2$

Step 8

Solve for $x$ .

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

Add $5 x$ to both sides of the equation.

$- 3 x + 8 + 5 x = 2$

Add $- 3 x$ and $5 x$ .

$2 x + 8 = 2$

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

Subtract $8$ from both sides of the equation.

$2 x = 2 - 8$

Subtract $8$ from $2$ .

$2 x = - 6$

Divide each term in $2 x = - 6$ by $2$ and simplify.

Divide each term in $2 x = - 6$ by $2$ .

$\frac{2 x}{2} = \frac{- 6}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x}{2} = \frac{- 6}{2}$

Divide $x$ by $1$ .

$x = \frac{- 6}{2}$

Simplify the right side.

Divide $- 6$ by $2$ .

$x = - 3$

Solve for x (5^(6x+5))/(125^(3x-1))=5^(-5x+2)

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