Solve for x (2^(3x-4))/(64^(x-5))=2^(8x-1)

$\frac{2^{3 x - 4}}{64^{x - 5}} = 2^{8 x - 1}$

Move $64^{x - 5}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

$2^{3 x - 4} \cdot 64^{- (x - 5)} = 2^{8 x - 1}$

Rewrite $64$ as $2^{6}$ .

$2^{3 x - 4} \cdot {(2^{6})}^{- (x - 5)} = 2^{8 x - 1}$

Multiply the exponents in ${(2^{6})}^{- (x - 5)}$ .

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

$2^{3 x - 4} \cdot 2^{6 (- (x - 5))} = 2^{8 x - 1}$

Apply the distributive property.

$2^{3 x - 4} \cdot 2^{6 (- x - - 5)} = 2^{8 x - 1}$

Multiply $- 1$ by $- 5$ .

$2^{3 x - 4} \cdot 2^{6 (- x + 5)} = 2^{8 x - 1}$

Apply the distributive property.

$2^{3 x - 4} \cdot 2^{6 (- x) + 6 \cdot 5} = 2^{8 x - 1}$

Multiply $- 1$ by $6$ .

$2^{3 x - 4} \cdot 2^{- 6 x + 6 \cdot 5} = 2^{8 x - 1}$

Multiply $6$ by $5$ .

$2^{3 x - 4} \cdot 2^{- 6 x + 30} = 2^{8 x - 1}$

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

$2^{3 x - 4 - 6 x + 30} = 2^{8 x - 1}$

Subtract $6 x$ from $3 x$ .

$2^{- 3 x - 4 + 30} = 2^{8 x - 1}$

Add $- 4$ and $30$ .

$2^{- 3 x + 26} = 2^{8 x - 1}$

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

$- 3 x + 26 = 8 x - 1$

Solve for $x$ .

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

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

$- 3 x + 26 - 8 x = - 1$

Subtract $8 x$ from $- 3 x$ .

$- 11 x + 26 = - 1$

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

Subtract $26$ from both sides of the equation.

$- 11 x = - 1 - 26$

Subtract $26$ from $- 1$ .

$- 11 x = - 27$

Divide each term in $- 11 x = - 27$ by $- 11$ and simplify.

Divide each term in $- 11 x = - 27$ by $- 11$ .

$\frac{- 11 x}{- 11} = \frac{- 27}{- 11}$

Simplify the left side.

Cancel the common factor of $- 11$ .

Cancel the common factor.

$\frac{- 11 x}{- 11} = \frac{- 27}{- 11}$

Divide $x$ by $1$ .

$x = \frac{- 27}{- 11}$

Simplify the right side.

Dividing two negative values results in a positive value.

$x = \frac{27}{11}$

The result can be shown in multiple forms.

Exact Form:

$x = \frac{27}{11}$

Decimal Form:

$x = 2.\overline{45}$

Mixed Number Form:

$x = 2 \frac{5}{11}$

Solve for x (2^(3x-4))/(64^(x-5))=2^(8x-1)

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