Algebra Examples

$(5^{\frac{x}{6}}) (5^{\frac{2 x}{3}}) = 5^{4}$

Step 1

Multiply $5^{\frac{x}{6}}$ by $5^{\frac{2 x}{3}}$ by adding the exponents.

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Step 1.1

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

$5^{\frac{x}{6} + \frac{2 x}{3}} = 5^{4}$

Step 1.2

To write $\frac{2 x}{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$5^{\frac{x}{6} + \frac{2 x}{3} \cdot \frac{2}{2}} = 5^{4}$

Step 1.3

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.3.1

Multiply $\frac{2 x}{3}$ by $\frac{2}{2}$ .

$5^{\frac{x}{6} + \frac{2 x \cdot 2}{3 \cdot 2}} = 5^{4}$

Step 1.3.2

Multiply $3$ by $2$ .

$5^{\frac{x}{6} + \frac{2 x \cdot 2}{6}} = 5^{4}$

Step 1.4

Combine the numerators over the common denominator.

$5^{\frac{x + 2 x \cdot 2}{6}} = 5^{4}$

Step 1.5

Simplify the numerator.

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Step 1.5.1

Factor $x$ out of $x + 2 x \cdot 2$ .

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Step 1.5.1.1

Raise $x$ to the power of $1$ .

$5^{\frac{x^{1} + 2 x \cdot 2}{6}} = 5^{4}$

Step 1.5.1.2

Factor $x$ out of $x^{1}$ .

$5^{\frac{x \cdot 1 + 2 x \cdot 2}{6}} = 5^{4}$

Step 1.5.1.3

Factor $x$ out of $2 x \cdot 2$ .

$5^{\frac{x \cdot 1 + x (2 \cdot 2)}{6}} = 5^{4}$

Step 1.5.1.4

Factor $x$ out of $x \cdot 1 + x (2 \cdot 2)$ .

$5^{\frac{x (1 + 2 \cdot 2)}{6}} = 5^{4}$

Step 1.5.2

Multiply $2$ by $2$ .

$5^{\frac{x (1 + 4)}{6}} = 5^{4}$

Step 1.5.3

Add $1$ and $4$ .

$5^{\frac{x \cdot 5}{6}} = 5^{4}$

Step 1.6

Move $5$ to the left of $x$ .

$5^{\frac{5 x}{6}} = 5^{4}$

Step 2

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

$\frac{5 x}{6} = 4$

Step 3

Solve for $x$ .

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Step 3.1

Multiply both sides of the equation by $\frac{6}{5}$ .

$\frac{6}{5} \cdot \frac{5 x}{6} = \frac{6}{5} \cdot 4$

Step 3.2

Simplify both sides of the equation.

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Step 3.2.1

Simplify the left side.

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Step 3.2.1.1

Simplify $\frac{6}{5} \cdot \frac{5 x}{6}$ .

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Step 3.2.1.1.1

Cancel the common factor of $6$ .

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Step 3.2.1.1.1.1

Cancel the common factor.

$\frac{6}{5} \cdot \frac{5 x}{6} = \frac{6}{5} \cdot 4$

Step 3.2.1.1.1.2

Rewrite the expression.

$\frac{1}{5} (5 x) = \frac{6}{5} \cdot 4$

Step 3.2.1.1.2

Cancel the common factor of $5$ .

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Step 3.2.1.1.2.1

Factor $5$ out of $5 x$ .

$\frac{1}{5} (5 (x)) = \frac{6}{5} \cdot 4$

Step 3.2.1.1.2.2

Cancel the common factor.

$\frac{1}{5} (5 x) = \frac{6}{5} \cdot 4$

Step 3.2.1.1.2.3

Rewrite the expression.

$x = \frac{6}{5} \cdot 4$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Multiply $\frac{6}{5} \cdot 4$ .

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Step 3.2.2.1.1

Combine $\frac{6}{5}$ and $4$ .

$x = \frac{6 \cdot 4}{5}$

Step 3.2.2.1.2

Multiply $6$ by $4$ .

$x = \frac{24}{5}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$x = \frac{24}{5}$

Decimal Form:

$x = 4.8$

Mixed Number Form:

$x = 4 \frac{4}{5}$