Algebra Examples

${(\sqrt[3]{6})}^{2 x} = {(\sqrt{6})}^{x + 6}$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{6}$ as $6^{\frac{1}{3}}$ .

${(6^{\frac{1}{3}})}^{2 x} = {\sqrt{6}}^{x + 6}$

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{6}$ as $6^{\frac{1}{2}}$ .

${(6^{\frac{1}{3}})}^{2 x} = {(6^{\frac{1}{2}})}^{x + 6}$

Step 3

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

$6^{\frac{1}{3} (2 x)} = {(6^{\frac{1}{2}})}^{x + 6}$

Step 4

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

$6^{\frac{1}{3} (2 x)} = 6^{\frac{1}{2} (x + 6)}$

Step 5

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

$\frac{1}{3} (2 x) = \frac{1}{2} (x + 6)$

Step 6

Solve for $x$ .

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Step 6.1

Multiply $\frac{1}{3} (2 x)$ .

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Step 6.1.1

Combine $2$ and $\frac{1}{3}$ .

$\frac{2}{3} x = \frac{1}{2} \cdot (x + 6)$

Step 6.1.2

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

$\frac{2 x}{3} = \frac{1}{2} \cdot (x + 6)$

Step 6.2

Simplify $\frac{1}{2} \cdot (x + 6)$ .

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Step 6.2.1

Apply the distributive property.

$\frac{2 x}{3} = \frac{1}{2} x + \frac{1}{2} \cdot 6$

Step 6.2.2

Combine $\frac{1}{2}$ and $x$ .

$\frac{2 x}{3} = \frac{x}{2} + \frac{1}{2} \cdot 6$

Step 6.2.3

Cancel the common factor of $2$ .

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Step 6.2.3.1

Factor $2$ out of $6$ .

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

Step 6.2.3.2

Cancel the common factor.

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

Step 6.2.3.3

Rewrite the expression.

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

Step 6.3

Move all terms containing $x$ to the left side of the equation.

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Step 6.3.1

Subtract $\frac{x}{2}$ from both sides of the equation.

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

Step 6.3.2

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

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

Step 6.3.3

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

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

Step 6.3.4

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

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Step 6.3.4.1

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

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

Step 6.3.4.2

Multiply $3$ by $2$ .

$\frac{2 x \cdot 2}{6} - \frac{x}{2} \cdot \frac{3}{3} = 3$

Step 6.3.4.3

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

$\frac{2 x \cdot 2}{6} - \frac{x \cdot 3}{2 \cdot 3} = 3$

Step 6.3.4.4

Multiply $2$ by $3$ .

$\frac{2 x \cdot 2}{6} - \frac{x \cdot 3}{6} = 3$

Step 6.3.5

Combine the numerators over the common denominator.

$\frac{2 x \cdot 2 - x \cdot 3}{6} = 3$

Step 6.3.6

Simplify the numerator.

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Step 6.3.6.1

Factor $x$ out of $2 x \cdot 2 - x \cdot 3$ .

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Step 6.3.6.1.1

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

$\frac{x (2 \cdot 2) - x \cdot 3}{6} = 3$

Step 6.3.6.1.2

Factor $x$ out of $- x \cdot 3$ .

$\frac{x (2 \cdot 2) + x (- 1 \cdot 3)}{6} = 3$

Step 6.3.6.1.3

Factor $x$ out of $x (2 \cdot 2) + x (- 1 \cdot 3)$ .

$\frac{x (2 \cdot 2 - 1 \cdot 3)}{6} = 3$

Step 6.3.6.2

Multiply $2$ by $2$ .

$\frac{x (4 - 1 \cdot 3)}{6} = 3$

Step 6.3.6.3

Multiply $- 1$ by $3$ .

$\frac{x (4 - 3)}{6} = 3$

Step 6.3.6.4

Subtract $3$ from $4$ .

$\frac{x \cdot 1}{6} = 3$

Step 6.3.7

Multiply $x$ by $1$ .

$\frac{x}{6} = 3$

Step 6.4

Multiply both sides of the equation by $6$ .

$6 \frac{x}{6} = 6 \cdot 3$

Step 6.5

Simplify both sides of the equation.

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Step 6.5.1

Simplify the left side.

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Step 6.5.1.1

Cancel the common factor of $6$ .

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Step 6.5.1.1.1

Cancel the common factor.

$6 \frac{x}{6} = 6 \cdot 3$

Step 6.5.1.1.2

Rewrite the expression.

$x = 6 \cdot 3$

Step 6.5.2

Simplify the right side.

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Step 6.5.2.1

Multiply $6$ by $3$ .

$x = 18$