Algebra Examples

$\sqrt[4]{83 x - 18} = 3 \sqrt[4]{x}$

Step 1

To remove the radical on the left side of the equation, raise both sides of the equation to the power of $4$ .

${\sqrt[4]{83 x - 18}}^{4} = {(3 \sqrt[4]{x})}^{4}$

Step 2

Simplify each side of the equation.

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Step 2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{83 x - 18}$ as ${(83 x - 18)}^{\frac{1}{4}}$ .

${({(83 x - 18)}^{\frac{1}{4}})}^{4} = {(3 \sqrt[4]{x})}^{4}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify ${({(83 x - 18)}^{\frac{1}{4}})}^{4}$ .

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Step 2.2.1.1

Multiply the exponents in ${({(83 x - 18)}^{\frac{1}{4}})}^{4}$ .

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Step 2.2.1.1.1

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

${(83 x - 18)}^{\frac{1}{4} \cdot 4} = {(3 \sqrt[4]{x})}^{4}$

Step 2.2.1.1.2

Cancel the common factor of $4$ .

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Step 2.2.1.1.2.1

Cancel the common factor.

${(83 x - 18)}^{\frac{1}{4} \cdot 4} = {(3 \sqrt[4]{x})}^{4}$

Step 2.2.1.1.2.2

Rewrite the expression.

${(83 x - 18)}^{1} = {(3 \sqrt[4]{x})}^{4}$

Step 2.2.1.2

Simplify.

$83 x - 18 = {(3 \sqrt[4]{x})}^{4}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Simplify ${(3 \sqrt[4]{x})}^{4}$ .

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Step 2.3.1.1

Simplify the expression.

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Step 2.3.1.1.1

Apply the product rule to $3 \sqrt[4]{x}$ .

$83 x - 18 = 3^{4} {\sqrt[4]{x}}^{4}$

Step 2.3.1.1.2

Raise $3$ to the power of $4$ .

$83 x - 18 = 81 {\sqrt[4]{x}}^{4}$

Step 2.3.1.2

Rewrite ${\sqrt[4]{x}}^{4}$ as $x$ .

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Step 2.3.1.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{x}$ as $x^{\frac{1}{4}}$ .

$83 x - 18 = 81 {(x^{\frac{1}{4}})}^{4}$

Step 2.3.1.2.2

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

$83 x - 18 = 81 x^{\frac{1}{4} \cdot 4}$

Step 2.3.1.2.3

Combine $\frac{1}{4}$ and $4$ .

$83 x - 18 = 81 x^{\frac{4}{4}}$

Step 2.3.1.2.4

Cancel the common factor of $4$ .

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Step 2.3.1.2.4.1

Cancel the common factor.

$83 x - 18 = 81 x^{\frac{4}{4}}$

Step 2.3.1.2.4.2

Rewrite the expression.

$83 x - 18 = 81 x^{1}$

Step 2.3.1.2.5

Simplify.

$83 x - 18 = 81 x$

Step 3

Solve for $x$ .

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Step 3.1

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

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Step 3.1.1

Subtract $81 x$ from both sides of the equation.

$83 x - 18 - 81 x = 0$

Step 3.1.2

Subtract $81 x$ from $83 x$ .

$2 x - 18 = 0$

Step 3.2

Add $18$ to both sides of the equation.

$2 x = 18$

Step 3.3

Divide each term in $2 x = 18$ by $2$ and simplify.

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Step 3.3.1

Divide each term in $2 x = 18$ by $2$ .

$\frac{2 x}{2} = \frac{18}{2}$

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Cancel the common factor of $2$ .

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Step 3.3.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{18}{2}$

Step 3.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{18}{2}$

Step 3.3.3

Simplify the right side.

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Step 3.3.3.1

Divide $18$ by $2$ .

$x = 9$