Algebra Examples

$\frac{1}{4} \sqrt[4]{2 x} + 6 = 10$

Step 1

Move all terms not containing $\frac{1}{4} \cdot \sqrt[4]{2 x}$ to the right side of the equation.

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Step 1.1

Subtract $6$ from both sides of the equation.

$\frac{1}{4} \cdot \sqrt[4]{2 x} = 10 - 6$

Step 1.2

Subtract $6$ from $10$ .

$\frac{1}{4} \cdot \sqrt[4]{2 x} = 4$

Step 2

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

${(\frac{1}{4} \cdot \sqrt[4]{2 x})}^{4} = 4^{4}$

Step 3

Simplify each side of the equation.

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Step 3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{2 x}$ as ${(2 x)}^{\frac{1}{4}}$ .

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

Step 3.2

Simplify the left side.

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Step 3.2.1

Simplify ${(\frac{1}{4} \cdot {(2 x)}^{\frac{1}{4}})}^{4}$ .

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Step 3.2.1.1

Apply the product rule to $2 x$ .

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

Step 3.2.1.2

Multiply $\frac{1}{4} (2^{\frac{1}{4}} x^{\frac{1}{4}})$ .

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Step 3.2.1.2.1

Combine $2^{\frac{1}{4}}$ and $\frac{1}{4}$ .

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

Step 3.2.1.2.2

Combine $\frac{2^{\frac{1}{4}}}{4}$ and $x^{\frac{1}{4}}$ .

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

Step 3.2.1.3

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 3.2.1.3.1

Apply the product rule to $\frac{2^{\frac{1}{4}} x^{\frac{1}{4}}}{4}$ .

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

Step 3.2.1.3.2

Apply the product rule to $2^{\frac{1}{4}} x^{\frac{1}{4}}$ .

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

Step 3.2.1.4

Simplify the numerator.

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Step 3.2.1.4.1

Multiply the exponents in ${(2^{\frac{1}{4}})}^{4}$ .

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Step 3.2.1.4.1.1

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

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

Step 3.2.1.4.1.2

Cancel the common factor of $4$ .

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Step 3.2.1.4.1.2.1

Cancel the common factor.

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

Step 3.2.1.4.1.2.2

Rewrite the expression.

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

Step 3.2.1.4.2

Evaluate the exponent.

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

Step 3.2.1.4.3

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

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Step 3.2.1.4.3.1

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

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

Step 3.2.1.4.3.2

Cancel the common factor of $4$ .

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Step 3.2.1.4.3.2.1

Cancel the common factor.

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

Step 3.2.1.4.3.2.2

Rewrite the expression.

$\frac{2 x^{1}}{4^{4}} = 4^{4}$

Step 3.2.1.4.4

Simplify.

$\frac{2 x}{4^{4}} = 4^{4}$

Step 3.2.1.5

Reduce the expression by cancelling the common factors.

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Step 3.2.1.5.1

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

$\frac{2 x}{256} = 4^{4}$

Step 3.2.1.5.2

Cancel the common factor of $2$ and $256$ .

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Step 3.2.1.5.2.1

Factor $2$ out of $2 x$ .

$\frac{2 (x)}{256} = 4^{4}$

Step 3.2.1.5.2.2

Cancel the common factors.

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Step 3.2.1.5.2.2.1

Factor $2$ out of $256$ .

$\frac{2 x}{2 \cdot 128} = 4^{4}$

Step 3.2.1.5.2.2.2

Cancel the common factor.

$\frac{2 x}{2 \cdot 128} = 4^{4}$

Step 3.2.1.5.2.2.3

Rewrite the expression.

$\frac{x}{128} = 4^{4}$

Step 3.3

Simplify the right side.

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Step 3.3.1

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

$\frac{x}{128} = 256$

Step 4

Solve for $x$ .

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Step 4.1

Multiply both sides of the equation by $128$ .

$128 \frac{x}{128} = 128 \cdot 256$

Step 4.2

Simplify both sides of the equation.

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Step 4.2.1

Simplify the left side.

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Step 4.2.1.1

Cancel the common factor of $128$ .

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Step 4.2.1.1.1

Cancel the common factor.

$128 \frac{x}{128} = 128 \cdot 256$

Step 4.2.1.1.2

Rewrite the expression.

$x = 128 \cdot 256$

Step 4.2.2

Simplify the right side.

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Step 4.2.2.1

Multiply $128$ by $256$ .

$x = 32768$