Algebra Examples

$3 {(x - 6)}^{4} + 11 = 15$

Step 1

Subtract $11$ from both sides of the equation.

$3 {(x - 6)}^{4} = 15 - 11$

Step 2

Subtract $11$ from $15$ .

$3 {(x - 6)}^{4} = 4$

Step 3

Divide each term in $3 {(x - 6)}^{4} = 4$ by $3$ and simplify.

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Step 3.1

Divide each term in $3 {(x - 6)}^{4} = 4$ by $3$ .

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

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $3$ .

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Step 3.2.1.1

Cancel the common factor.

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

Step 3.2.1.2

Divide ${(x - 6)}^{4}$ by $1$ .

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

Step 4

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x - 6 = \pm \sqrt[4]{\frac{4}{3}}$

Step 5

Simplify $\pm \sqrt[4]{\frac{4}{3}}$ .

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Step 5.1

Rewrite $\sqrt[4]{\frac{4}{3}}$ as $\frac{\sqrt[4]{4}}{\sqrt[4]{3}}$ .

$x - 6 = \pm \frac{\sqrt[4]{4}}{\sqrt[4]{3}}$

Step 5.2

Simplify the numerator.

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Step 5.2.1

Rewrite $4$ as $2^{2}$ .

$x - 6 = \pm \frac{\sqrt[4]{2^{2}}}{\sqrt[4]{3}}$

Step 5.2.2

Rewrite $\sqrt[4]{2^{2}}$ as $\sqrt{\sqrt{2^{2}}}$ .

$x - 6 = \pm \frac{\sqrt{\sqrt{2^{2}}}}{\sqrt[4]{3}}$

Step 5.2.3

Pull terms out from under the radical, assuming positive real numbers.

$x - 6 = \pm \frac{\sqrt{2}}{\sqrt[4]{3}}$

Step 5.3

Multiply $\frac{\sqrt{2}}{\sqrt[4]{3}}$ by $\frac{{\sqrt[4]{3}}^{3}}{{\sqrt[4]{3}}^{3}}$ .

$x - 6 = \pm \frac{\sqrt{2}}{\sqrt[4]{3}} \cdot \frac{{\sqrt[4]{3}}^{3}}{{\sqrt[4]{3}}^{3}}$

Step 5.4

Combine and simplify the denominator.

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Step 5.4.1

Multiply $\frac{\sqrt{2}}{\sqrt[4]{3}}$ by $\frac{{\sqrt[4]{3}}^{3}}{{\sqrt[4]{3}}^{3}}$ .

$x - 6 = \pm \frac{\sqrt{2} {\sqrt[4]{3}}^{3}}{\sqrt[4]{3} {\sqrt[4]{3}}^{3}}$

Step 5.4.2

Raise $\sqrt[4]{3}$ to the power of $1$ .

$x - 6 = \pm \frac{\sqrt{2} {\sqrt[4]{3}}^{3}}{{\sqrt[4]{3}}^{1} {\sqrt[4]{3}}^{3}}$

Step 5.4.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x - 6 = \pm \frac{\sqrt{2} {\sqrt[4]{3}}^{3}}{{\sqrt[4]{3}}^{1 + 3}}$

Step 5.4.4

Add $1$ and $3$ .

$x - 6 = \pm \frac{\sqrt{2} {\sqrt[4]{3}}^{3}}{{\sqrt[4]{3}}^{4}}$

Step 5.4.5

Rewrite ${\sqrt[4]{3}}^{4}$ as $3$ .

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Step 5.4.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{3}$ as $3^{\frac{1}{4}}$ .

$x - 6 = \pm \frac{\sqrt{2} {\sqrt[4]{3}}^{3}}{{(3^{\frac{1}{4}})}^{4}}$

Step 5.4.5.2

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

$x - 6 = \pm \frac{\sqrt{2} {\sqrt[4]{3}}^{3}}{3^{\frac{1}{4} \cdot 4}}$

Step 5.4.5.3

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

$x - 6 = \pm \frac{\sqrt{2} {\sqrt[4]{3}}^{3}}{3^{\frac{4}{4}}}$

Step 5.4.5.4

Cancel the common factor of $4$ .

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Step 5.4.5.4.1

Cancel the common factor.

$x - 6 = \pm \frac{\sqrt{2} {\sqrt[4]{3}}^{3}}{3^{\frac{4}{4}}}$

Step 5.4.5.4.2

Rewrite the expression.

$x - 6 = \pm \frac{\sqrt{2} {\sqrt[4]{3}}^{3}}{3^{1}}$

Step 5.4.5.5

Evaluate the exponent.

$x - 6 = \pm \frac{\sqrt{2} {\sqrt[4]{3}}^{3}}{3}$

Step 5.5

Simplify the numerator.

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Step 5.5.1

Rewrite ${\sqrt[4]{3}}^{3}$ as $\sqrt[4]{3^{3}}$ .

$x - 6 = \pm \frac{\sqrt{2} \sqrt[4]{3^{3}}}{3}$

Step 5.5.2

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

$x - 6 = \pm \frac{\sqrt{2} \sqrt[4]{27}}{3}$

Step 5.6

Simplify the numerator.

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Step 5.6.1

Rewrite the expression using the least common index of $4$ .

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Step 5.6.1.1

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

$x - 6 = \pm \frac{2^{\frac{1}{2}} \sqrt[4]{27}}{3}$

Step 5.6.1.2

Rewrite $2^{\frac{1}{2}}$ as $2^{\frac{2}{4}}$ .

$x - 6 = \pm \frac{2^{\frac{2}{4}} \sqrt[4]{27}}{3}$

Step 5.6.1.3

Rewrite $2^{\frac{2}{4}}$ as $\sqrt[4]{2^{2}}$ .

$x - 6 = \pm \frac{\sqrt[4]{2^{2}} \sqrt[4]{27}}{3}$

Step 5.6.2

Combine using the product rule for radicals.

$x - 6 = \pm \frac{\sqrt[4]{2^{2} \cdot 27}}{3}$

Step 5.6.3

Raise $2$ to the power of $2$ .

$x - 6 = \pm \frac{\sqrt[4]{4 \cdot 27}}{3}$

Step 5.7

Multiply $4$ by $27$ .

$x - 6 = \pm \frac{\sqrt[4]{108}}{3}$

Step 6

The complete solution is the result of both the positive and negative portions of the solution.

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Step 6.1

First, use the positive value of the $\pm$ to find the first solution.

$x - 6 = \frac{\sqrt[4]{108}}{3}$

Step 6.2

Add $6$ to both sides of the equation.

$x = \frac{\sqrt[4]{108}}{3} + 6$

Step 6.3

Next, use the negative value of the $\pm$ to find the second solution.

$x - 6 = - \frac{\sqrt[4]{108}}{3}$

Step 6.4

Add $6$ to both sides of the equation.

$x = - \frac{\sqrt[4]{108}}{3} + 6$

Step 6.5

The complete solution is the result of both the positive and negative portions of the solution.

$x = \frac{\sqrt[4]{108}}{3} + 6, - \frac{\sqrt[4]{108}}{3} + 6$

Step 7

The result can be shown in multiple forms.

Exact Form:

$x = \frac{\sqrt[4]{108}}{3} + 6, - \frac{\sqrt[4]{108}}{3} + 6$

Decimal Form:

$x = 7.07456993 \dots, 4.92543006 \dots$