Algebra Examples

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

Step 1

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

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

Step 1.2

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

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

Step 1.3

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

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

Step 1.4

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

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Step 1.4.1

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

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

Step 1.4.2

Multiply $3$ by $2$ .

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

Step 1.4.3

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

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

Step 1.4.4

Multiply $2$ by $3$ .

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

Step 1.5

Combine the numerators over the common denominator.

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

Step 1.6

Simplify the numerator.

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Step 1.6.1

Move $2$ to the left of $x$ .

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

Step 1.6.2

Move $3$ to the left of $x$ .

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

Step 1.6.3

Add $2 x$ and $3 x$ .

$2^{\frac{5 x}{6}} = 2^{5}$

Step 2

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

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

Step 3

Solve for $x$ .

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Step 3.1

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

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

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.

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

Step 3.2.1.1.1.2

Rewrite the expression.

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

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 $5 x$ .

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

Step 3.2.1.1.2.2

Cancel the common factor.

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

Step 3.2.1.1.2.3

Rewrite the expression.

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

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Cancel the common factor of $5$ .

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Step 3.2.2.1.1

Cancel the common factor.

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

Step 3.2.2.1.2

Rewrite the expression.

$x = 6$