Solve for x 4^(3x-1)*(1/8)^x=32^(-x)

$4^{3 x - 1} \cdot {(\frac{1}{8})}^{x} = 32^{- x}$

Apply the product rule to $\frac{1}{8}$ .

$4^{3 x - 1} \cdot \frac{1^{x}}{8^{x}} = 32^{- x}$

One to any power is one.

$4^{3 x - 1} \cdot \frac{1}{8^{x}} = 32^{- x}$

Move $8^{x}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

$4^{3 x - 1} \cdot 8^{- x} = 32^{- x}$

Rewrite $4$ as $2^{2}$ .

${(2^{2})}^{3 x - 1} \cdot 8^{- x} = 32^{- x}$

Multiply the exponents in ${(2^{2})}^{3 x - 1}$ .

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

$2^{2 (3 x - 1)} \cdot 8^{- x} = 32^{- x}$

Apply the distributive property.

$2^{2 (3 x) + 2 \cdot - 1} \cdot 8^{- x} = 32^{- x}$

Multiply $3$ by $2$ .

$2^{6 x + 2 \cdot - 1} \cdot 8^{- x} = 32^{- x}$

Multiply $2$ by $- 1$ .

$2^{6 x - 2} \cdot 8^{- x} = 32^{- x}$

Rewrite $8$ as $2^{3}$ .

$2^{6 x - 2} \cdot {(2^{3})}^{- x} = 32^{- x}$

Multiply the exponents in ${(2^{3})}^{- x}$ .

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

$2^{6 x - 2} \cdot 2^{3 (- x)} = 32^{- x}$

Multiply $- 1$ by $3$ .

$2^{6 x - 2} \cdot 2^{- 3 x} = 32^{- x}$

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

$2^{6 x - 2 - 3 x} = 32^{- x}$

Subtract $3 x$ from $6 x$ .

$2^{3 x - 2} = 32^{- x}$

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

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

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

$3 x - 2 = 5 (- x)$

Solve for $x$ .

Multiply $- 1$ by $5$ .

$3 x - 2 = - 5 x$

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

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

$3 x - 2 + 5 x = 0$

Add $3 x$ and $5 x$ .

$8 x - 2 = 0$

Add $2$ to both sides of the equation.

$8 x = 2$

Divide each term in $8 x = 2$ by $8$ and simplify.

Divide each term in $8 x = 2$ by $8$ .

$\frac{8 x}{8} = \frac{2}{8}$

Simplify the left side.

Cancel the common factor of $8$ .

Cancel the common factor.

$\frac{8 x}{8} = \frac{2}{8}$

Divide $x$ by $1$ .

$x = \frac{2}{8}$

Simplify the right side.

Cancel the common factor of $2$ and $8$ .

Factor $2$ out of $2$ .

$x = \frac{2 (1)}{8}$

Cancel the common factors.

Factor $2$ out of $8$ .

$x = \frac{2 \cdot 1}{2 \cdot 4}$

Cancel the common factor.

$x = \frac{2 \cdot 1}{2 \cdot 4}$

Rewrite the expression.

$x = \frac{1}{4}$

The result can be shown in multiple forms.

Exact Form:

$x = \frac{1}{4}$

Decimal Form:

$x = 0.25$

Solve for x 4^(3x-1)*(1/8)^x=32^(-x)

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