Algebra Examples

$32 {(\frac{1}{4})}^{\frac{x}{3}} = 2$

Step 1

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

$32 \frac{1^{\frac{x}{3}}}{4^{\frac{x}{3}}} = 2$

Step 2

One to any power is one.

$32 \frac{1}{4^{\frac{x}{3}}} = 2$

Step 3

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

$32 \cdot 4^{- \frac{x}{3}} = 2$

Step 4

Rewrite $32$ as $2^{5}$ .

$2^{5} \cdot 4^{- \frac{x}{3}} = 2$

Step 5

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

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

Step 6

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

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Step 6.1

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

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

Step 6.2

Multiply $2 (- \frac{x}{3})$ .

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Step 6.2.1

Multiply $- 1$ by $2$ .

$2^{5} \cdot 2^{- 2 \frac{x}{3}} = 2$

Step 6.2.2

Combine $- 2$ and $\frac{x}{3}$ .

$2^{5} \cdot 2^{\frac{- 2 x}{3}} = 2$

Step 6.3

Move the negative in front of the fraction.

$2^{5} \cdot 2^{- \frac{2 x}{3}} = 2$

Step 7

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

$2^{5 - \frac{2 x}{3}} = 2$

Step 8

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

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

Step 9

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

$5 - \frac{2 x}{3} = 1$

Step 10

Solve for $x$ .

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Step 10.1

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

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Step 10.1.1

Subtract $5$ from both sides of the equation.

$- \frac{2 x}{3} = 1 - 5$

Step 10.1.2

Subtract $5$ from $1$ .

$- \frac{2 x}{3} = - 4$

Step 10.2

Multiply both sides of the equation by $- \frac{3}{2}$ .

$- \frac{3}{2} (- \frac{2 x}{3}) = - \frac{3}{2} \cdot - 4$

Step 10.3

Simplify both sides of the equation.

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Step 10.3.1

Simplify the left side.

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Step 10.3.1.1

Simplify $- \frac{3}{2} (- \frac{2 x}{3})$ .

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Step 10.3.1.1.1

Cancel the common factor of $3$ .

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Step 10.3.1.1.1.1

Move the leading negative in $- \frac{3}{2}$ into the numerator.

$\frac{- 3}{2} (- \frac{2 x}{3}) = - \frac{3}{2} \cdot - 4$

Step 10.3.1.1.1.2

Move the leading negative in $- \frac{2 x}{3}$ into the numerator.

$\frac{- 3}{2} \cdot \frac{- 2 x}{3} = - \frac{3}{2} \cdot - 4$

Step 10.3.1.1.1.3

Factor $3$ out of $- 3$ .

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

Step 10.3.1.1.1.4

Cancel the common factor.

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

Step 10.3.1.1.1.5

Rewrite the expression.

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

Step 10.3.1.1.2

Cancel the common factor of $2$ .

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Step 10.3.1.1.2.1

Factor $2$ out of $- 2 x$ .

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

Step 10.3.1.1.2.2

Cancel the common factor.

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

Step 10.3.1.1.2.3

Rewrite the expression.

$- - x = - \frac{3}{2} \cdot - 4$

Step 10.3.1.1.3

Multiply.

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Step 10.3.1.1.3.1

Multiply $- 1$ by $- 1$ .

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

Step 10.3.1.1.3.2

Multiply $x$ by $1$ .

$x = - \frac{3}{2} \cdot - 4$

Step 10.3.2

Simplify the right side.

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Step 10.3.2.1

Simplify $- \frac{3}{2} \cdot - 4$ .

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Step 10.3.2.1.1

Cancel the common factor of $2$ .

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Step 10.3.2.1.1.1

Move the leading negative in $- \frac{3}{2}$ into the numerator.

$x = \frac{- 3}{2} \cdot - 4$

Step 10.3.2.1.1.2

Factor $2$ out of $- 4$ .

$x = \frac{- 3}{2} \cdot (2 (- 2))$

Step 10.3.2.1.1.3

Cancel the common factor.

$x = \frac{- 3}{2} \cdot (2 \cdot - 2)$

Step 10.3.2.1.1.4

Rewrite the expression.

$x = - 3 \cdot - 2$

Step 10.3.2.1.2

Multiply $- 3$ by $- 2$ .

$x = 6$