Solve for x (49^(4x-1))/(7^(x-6))=7^(8x+3)

$\frac{49^{4 x - 1}}{7^{x - 6}} = 7^{8 x + 3}$

Step 1

Move $7^{x - 6}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

$49^{4 x - 1} \cdot 7^{- (x - 6)} = 7^{8 x + 3}$

Step 2

Rewrite $49$ as $7^{2}$ .

${(7^{2})}^{4 x - 1} \cdot 7^{- (x - 6)} = 7^{8 x + 3}$

Step 3

Multiply the exponents in ${(7^{2})}^{4 x - 1}$ .

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

$7^{2 (4 x - 1)} \cdot 7^{- (x - 6)} = 7^{8 x + 3}$

Apply the distributive property.

$7^{2 (4 x) + 2 \cdot - 1} \cdot 7^{- (x - 6)} = 7^{8 x + 3}$

Multiply $4$ by $2$ .

$7^{8 x + 2 \cdot - 1} \cdot 7^{- (x - 6)} = 7^{8 x + 3}$

Multiply $2$ by $- 1$ .

$7^{8 x - 2} \cdot 7^{- (x - 6)} = 7^{8 x + 3}$

Step 4

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

$7^{8 x - 2 - (x - 6)} = 7^{8 x + 3}$

Step 5

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

$8 x - 2 - (x - 6) = 8 x + 3$

Step 6

Solve for $x$ .

Simplify $8 x - 2 - (x - 6)$ .

Simplify each term.

Apply the distributive property.

$8 x - 2 - x - - 6 = 8 x + 3$

Multiply $- 1$ by $- 6$ .

$8 x - 2 - x + 6 = 8 x + 3$

Simplify by adding terms.

Subtract $x$ from $8 x$ .

$7 x - 2 + 6 = 8 x + 3$

Add $- 2$ and $6$ .

$7 x + 4 = 8 x + 3$

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

Subtract $8 x$ from both sides of the equation.

$7 x + 4 - 8 x = 3$

Subtract $8 x$ from $7 x$ .

$- x + 4 = 3$

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

Subtract $4$ from both sides of the equation.

$- x = 3 - 4$

Subtract $4$ from $3$ .

$- x = - 1$

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

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

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

Simplify the left side.

Dividing two negative values results in a positive value.

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

Divide $x$ by $1$ .

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

Simplify the right side.

Divide $- 1$ by $- 1$ .

$x = 1$

Solve for x (49^(4x-1))/(7^(x-6))=7^(8x+3)

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