Precalculus Examples

$36 {(\frac{1}{3})}^{\frac{x}{5}} = 4$

Step 1

Divide each term in $36 {(\frac{1}{3})}^{\frac{x}{5}} = 4$ by $36$ and simplify.

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Step 1.1

Divide each term in $36 {(\frac{1}{3})}^{\frac{x}{5}} = 4$ by $36$ .

$\frac{36 {(\frac{1}{3})}^{\frac{x}{5}}}{36} = \frac{4}{36}$

Step 1.2

Simplify the left side.

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Step 1.2.1

Cancel the common factor of $36$ .

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Step 1.2.1.1

Cancel the common factor.

$\frac{36 {(\frac{1}{3})}^{\frac{x}{5}}}{36} = \frac{4}{36}$

Step 1.2.1.2

Divide ${(\frac{1}{3})}^{\frac{x}{5}}$ by $1$ .

${(\frac{1}{3})}^{\frac{x}{5}} = \frac{4}{36}$

Step 1.2.2

Simplify the expression.

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Step 1.2.2.1

Apply the product rule to $\frac{1}{3}$ .

$\frac{1^{\frac{x}{5}}}{3^{\frac{x}{5}}} = \frac{4}{36}$

Step 1.2.2.2

One to any power is one.

$\frac{1}{3^{\frac{x}{5}}} = \frac{4}{36}$

Step 1.3

Simplify the right side.

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Step 1.3.1

Cancel the common factor of $4$ and $36$ .

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Step 1.3.1.1

Factor $4$ out of $4$ .

$\frac{1}{3^{\frac{x}{5}}} = \frac{4 (1)}{36}$

Step 1.3.1.2

Cancel the common factors.

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Step 1.3.1.2.1

Factor $4$ out of $36$ .

$\frac{1}{3^{\frac{x}{5}}} = \frac{4 \cdot 1}{4 \cdot 9}$

Step 1.3.1.2.2

Cancel the common factor.

$\frac{1}{3^{\frac{x}{5}}} = \frac{4 \cdot 1}{4 \cdot 9}$

Step 1.3.1.2.3

Rewrite the expression.

$\frac{1}{3^{\frac{x}{5}}} = \frac{1}{9}$

Step 2

Multiply both sides by $3^{\frac{x}{5}}$ .

$\frac{1}{3^{\frac{x}{5}}} \cdot 3^{\frac{x}{5}} = \frac{1}{9} \cdot 3^{\frac{x}{5}}$

Step 3

Simplify.

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Step 3.1

Simplify the left side.

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Step 3.1.1

Cancel the common factor of $3^{\frac{x}{5}}$ .

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Step 3.1.1.1

Cancel the common factor.

$\frac{1}{3^{\frac{x}{5}}} \cdot 3^{\frac{x}{5}} = \frac{1}{9} \cdot 3^{\frac{x}{5}}$

Step 3.1.1.2

Rewrite the expression.

$1 = \frac{1}{9} \cdot 3^{\frac{x}{5}}$

Step 3.2

Simplify the right side.

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Step 3.2.1

Combine $\frac{1}{9}$ and $3^{\frac{x}{5}}$ .

$1 = \frac{3^{\frac{x}{5}}}{9}$

Step 4

Solve for $x$ .

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Step 4.1

Rewrite the equation as $\frac{3^{\frac{x}{5}}}{9} = 1$ .

$\frac{3^{\frac{x}{5}}}{9} = 1$

Step 4.2

Multiply both sides of the equation by $9$ .

$9 \frac{3^{\frac{x}{5}}}{9} = 9 \cdot 1$

Step 4.3

Simplify both sides of the equation.

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Step 4.3.1

Simplify the left side.

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Step 4.3.1.1

Cancel the common factor of $9$ .

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Step 4.3.1.1.1

Cancel the common factor.

$9 \frac{3^{\frac{x}{5}}}{9} = 9 \cdot 1$

Step 4.3.1.1.2

Rewrite the expression.

$3^{\frac{x}{5}} = 9 \cdot 1$

Step 4.3.2

Simplify the right side.

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Step 4.3.2.1

Multiply $9$ by $1$ .

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

Step 4.4

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

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

Step 4.5

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

$\frac{x}{5} = 2$

Step 4.6

Solve for $x$ .

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Step 4.6.1

Multiply both sides of the equation by $5$ .

$5 \frac{x}{5} = 5 \cdot 2$

Step 4.6.2

Simplify both sides of the equation.

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Step 4.6.2.1

Simplify the left side.

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Step 4.6.2.1.1

Cancel the common factor of $5$ .

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Step 4.6.2.1.1.1

Cancel the common factor.

$5 \frac{x}{5} = 5 \cdot 2$

Step 4.6.2.1.1.2

Rewrite the expression.

$x = 5 \cdot 2$

Step 4.6.2.2

Simplify the right side.

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Step 4.6.2.2.1

Multiply $5$ by $2$ .

$x = 10$