Algebra Examples

${(\sqrt[5]{3})}^{5 x - 10} = {(\sqrt[8]{3})}^{4 x}$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{3}$ as $3^{\frac{1}{5}}$ .

${(3^{\frac{1}{5}})}^{5 x - 10} = {\sqrt[8]{3}}^{4 x}$

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[8]{3}$ as $3^{\frac{1}{8}}$ .

${(3^{\frac{1}{5}})}^{5 x - 10} = {(3^{\frac{1}{8}})}^{4 x}$

Step 3

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

$3^{\frac{1}{5} (5 x - 10)} = {(3^{\frac{1}{8}})}^{4 x}$

Step 4

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

$3^{\frac{1}{5} (5 x - 10)} = 3^{\frac{1}{8} (4 x)}$

Step 5

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

$\frac{1}{5} (5 x - 10) = \frac{1}{8} (4 x)$

Step 6

Solve for $x$ .

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Step 6.1

Simplify $\frac{1}{5} \cdot (5 x - 10)$ .

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Step 6.1.1

Rewrite.

$0 + 0 + \frac{1}{5} \cdot (5 x - 10) = \frac{1}{8} \cdot (4 x)$

Step 6.1.2

Simplify by adding zeros.

$\frac{1}{5} \cdot (5 x - 10) = \frac{1}{8} \cdot (4 x)$

Step 6.1.3

Apply the distributive property.

$\frac{1}{5} (5 x) + \frac{1}{5} \cdot - 10 = \frac{1}{8} \cdot (4 x)$

Step 6.1.4

Cancel the common factor of $5$ .

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Step 6.1.4.1

Factor $5$ out of $5 x$ .

$\frac{1}{5} (5 (x)) + \frac{1}{5} \cdot - 10 = \frac{1}{8} \cdot (4 x)$

Step 6.1.4.2

Cancel the common factor.

$\frac{1}{5} (5 x) + \frac{1}{5} \cdot - 10 = \frac{1}{8} \cdot (4 x)$

Step 6.1.4.3

Rewrite the expression.

$x + \frac{1}{5} \cdot - 10 = \frac{1}{8} \cdot (4 x)$

Step 6.1.5

Cancel the common factor of $5$ .

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Step 6.1.5.1

Factor $5$ out of $- 10$ .

$x + \frac{1}{5} \cdot (5 (- 2)) = \frac{1}{8} \cdot (4 x)$

Step 6.1.5.2

Cancel the common factor.

$x + \frac{1}{5} \cdot (5 \cdot - 2) = \frac{1}{8} \cdot (4 x)$

Step 6.1.5.3

Rewrite the expression.

$x - 2 = \frac{1}{8} \cdot (4 x)$

Step 6.2

Simplify $\frac{1}{8} \cdot (4 x)$ .

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Step 6.2.1

Cancel the common factor of $4$ .

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Step 6.2.1.1

Factor $4$ out of $8$ .

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

Step 6.2.1.2

Factor $4$ out of $4 x$ .

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

Step 6.2.1.3

Cancel the common factor.

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

Step 6.2.1.4

Rewrite the expression.

$x - 2 = \frac{1}{2} \cdot x$

Step 6.2.2

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

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

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.

$x - 2 - \frac{x}{2} = 0$

Step 6.3.2

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

$x \cdot \frac{2}{2} - \frac{x}{2} - 2 = 0$

Step 6.3.3

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

$\frac{x \cdot 2}{2} - \frac{x}{2} - 2 = 0$

Step 6.3.4

Combine the numerators over the common denominator.

$\frac{x \cdot 2 - x}{2} - 2 = 0$

Step 6.3.5

Subtract $x$ from $x \cdot 2$ .

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Step 6.3.5.1

Reorder $x$ and $2$ .

$\frac{2 \cdot x - x}{2} - 2 = 0$

Step 6.3.5.2

Subtract $x$ from $2 \cdot x$ .

$\frac{x}{2} - 2 = 0$

Step 6.4

Add $2$ to both sides of the equation.

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

Step 6.5

Multiply both sides of the equation by $2$ .

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

Step 6.6

Simplify both sides of the equation.

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Step 6.6.1

Simplify the left side.

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Step 6.6.1.1

Cancel the common factor of $2$ .

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Step 6.6.1.1.1

Cancel the common factor.

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

Step 6.6.1.1.2

Rewrite the expression.

$x = 2 \cdot 2$

Step 6.6.2

Simplify the right side.

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Step 6.6.2.1

Multiply $2$ by $2$ .

$x = 4$