Algebra Examples

${(\frac{1}{4})}^{3 - 2 x} = 8^{- x}$

Step 1

Apply the product rule to $\frac{1}{4}$ .

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

Step 2

One to any power is one.

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

Step 3

Move $4^{3 - 2 x}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

$4^{- (3 - 2 x)} = 8^{- x}$

Step 4

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

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

Step 5

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

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

Step 6

Solve for $x$ .

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Step 6.1

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

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Step 6.1.1

Rewrite.

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

Step 6.1.2

Simplify by adding zeros.

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

Step 6.1.3

Apply the distributive property.

$2 (- 1 \cdot 3 - (- 2 x)) = 3 (- x)$

Step 6.1.4

Multiply.

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Step 6.1.4.1

Multiply $- 1$ by $3$ .

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

Step 6.1.4.2

Multiply $- 2$ by $- 1$ .

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

Step 6.1.5

Apply the distributive property.

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

Step 6.1.6

Multiply.

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Step 6.1.6.1

Multiply $2$ by $- 3$ .

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

Step 6.1.6.2

Multiply $2$ by $2$ .

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

Step 6.2

Multiply $- 1$ by $3$ .

$- 6 + 4 x = - 3 x$

Step 6.3

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

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Step 6.3.1

Add $3 x$ to both sides of the equation.

$- 6 + 4 x + 3 x = 0$

Step 6.3.2

Add $4 x$ and $3 x$ .

$- 6 + 7 x = 0$

Step 6.4

Add $6$ to both sides of the equation.

$7 x = 6$

Step 6.5

Divide each term in $7 x = 6$ by $7$ and simplify.

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Step 6.5.1

Divide each term in $7 x = 6$ by $7$ .

$\frac{7 x}{7} = \frac{6}{7}$

Step 6.5.2

Simplify the left side.

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Step 6.5.2.1

Cancel the common factor of $7$ .

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Step 6.5.2.1.1

Cancel the common factor.

$\frac{7 x}{7} = \frac{6}{7}$

Step 6.5.2.1.2

Divide $x$ by $1$ .

$x = \frac{6}{7}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$x = \frac{6}{7}$

Decimal Form:

$x = 0.\overline{857142}$