Algebra Examples

$256 = {(\frac{1}{4})}^{3 x + 2}$

Step 1

Rewrite the equation as ${(\frac{1}{4})}^{3 x + 2} = 256$ .

${(\frac{1}{4})}^{3 x + 2} = 256$

Step 2

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

$\frac{1^{3 x + 2}}{4^{3 x + 2}} = 256$

Step 3

One to any power is one.

$\frac{1}{4^{3 x + 2}} = 256$

Step 4

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

$4^{- (3 x + 2)} = 256$

Step 5

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

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

Step 6

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

$- (3 x + 2) = 4$

Step 7

Solve for $x$ .

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Step 7.1

Simplify $- (3 x + 2)$ .

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Step 7.1.1

Apply the distributive property.

$- (3 x) - 1 \cdot 2 = 4$

Step 7.1.2

Multiply.

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Step 7.1.2.1

Multiply $3$ by $- 1$ .

$- 3 x - 1 \cdot 2 = 4$

Step 7.1.2.2

Multiply $- 1$ by $2$ .

$- 3 x - 2 = 4$

Step 7.2

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

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Step 7.2.1

Add $2$ to both sides of the equation.

$- 3 x = 4 + 2$

Step 7.2.2

Add $4$ and $2$ .

$- 3 x = 6$

Step 7.3

Divide each term in $- 3 x = 6$ by $- 3$ and simplify.

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Step 7.3.1

Divide each term in $- 3 x = 6$ by $- 3$ .

$\frac{- 3 x}{- 3} = \frac{6}{- 3}$

Step 7.3.2

Simplify the left side.

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Step 7.3.2.1

Cancel the common factor of $- 3$ .

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Step 7.3.2.1.1

Cancel the common factor.

$\frac{- 3 x}{- 3} = \frac{6}{- 3}$

Step 7.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{6}{- 3}$

Step 7.3.3

Simplify the right side.

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Step 7.3.3.1

Divide $6$ by $- 3$ .

$x = - 2$