Precalculus Examples

$32 \cdot (4^{\frac{x}{3}}) = 512$

Step 1

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

$2^{5} \cdot 4^{\frac{x}{3}} = 512$

Step 2

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

$2^{5} \cdot {(2^{2})}^{\frac{x}{3}} = 512$

Step 3

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

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Step 3.1

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

$2^{5} \cdot 2^{2 \frac{x}{3}} = 512$

Step 3.2

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

$2^{5} \cdot 2^{\frac{2 x}{3}} = 512$

Step 4

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

$2^{5 + \frac{2 x}{3}} = 512$

Step 5

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

$2^{5 + \frac{2 x}{3}} = 2^{9}$

Step 6

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

$5 + \frac{2 x}{3} = 9$

Step 7

Solve for $x$ .

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Step 7.1

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

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Step 7.1.1

Subtract $5$ from both sides of the equation.

$\frac{2 x}{3} = 9 - 5$

Step 7.1.2

Subtract $5$ from $9$ .

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

Step 7.2

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

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

Step 7.3

Simplify both sides of the equation.

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Step 7.3.1

Simplify the left side.

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Step 7.3.1.1

Simplify $\frac{3}{2} \cdot \frac{2 x}{3}$ .

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Step 7.3.1.1.1

Cancel the common factor of $3$ .

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Step 7.3.1.1.1.1

Cancel the common factor.

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

Step 7.3.1.1.1.2

Rewrite the expression.

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

Step 7.3.1.1.2

Cancel the common factor of $2$ .

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Step 7.3.1.1.2.1

Factor $2$ out of $2 x$ .

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

Step 7.3.1.1.2.2

Cancel the common factor.

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

Step 7.3.1.1.2.3

Rewrite the expression.

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

Step 7.3.2

Simplify the right side.

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Step 7.3.2.1

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

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Step 7.3.2.1.1

Cancel the common factor of $2$ .

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Step 7.3.2.1.1.1

Factor $2$ out of $4$ .

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

Step 7.3.2.1.1.2

Cancel the common factor.

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

Step 7.3.2.1.1.3

Rewrite the expression.

$x = 3 \cdot 2$

Step 7.3.2.1.2

Multiply $3$ by $2$ .

$x = 6$