Algebra Examples

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

Step 1

Rewrite the equation as $4 - 5 \sqrt[3]{x^{8}} = 3$ .

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

Step 2

Simplify each term.

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Step 2.1

Rewrite $x^{8}$ as ${(x^{2})}^{3} x^{2}$ .

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Step 2.1.1

Factor out $x^{6}$ .

$4 - 5 \sqrt[3]{x^{6} x^{2}} = 3$

Step 2.1.2

Rewrite $x^{6}$ as ${(x^{2})}^{3}$ .

$4 - 5 \sqrt[3]{{(x^{2})}^{3} x^{2}} = 3$

Step 2.2

Pull terms out from under the radical.

$4 - 5 x^{2} \sqrt[3]{x^{2}} = 3$

Step 3

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{x^{2}}$ as $x^{\frac{2}{3}}$ .

$4 - 5 x^{2} x^{\frac{2}{3}} = 3$

Step 4

Move the terms containing $x$ to the left side and simplify.

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Step 4.1

Move all terms not containing $x$ to the right side of the equation.

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Step 4.1.1

Subtract $4$ from both sides of the equation.

$- 5 x^{2} x^{\frac{2}{3}} - 3 = - 4$

Step 4.1.2

Add $3$ to both sides of the equation.

$- 5 x^{2} x^{\frac{2}{3}} = - 4 + 3$

Step 4.1.3

Add $- 4$ and $3$ .

$- 5 x^{2} x^{\frac{2}{3}} = - 1$

Step 4.2

Multiply $x^{2}$ by $x^{\frac{2}{3}}$ by adding the exponents.

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Step 4.2.1

Move $x^{\frac{2}{3}}$ .

$- 5 (x^{\frac{2}{3}} x^{2}) = - 1$

Step 4.2.2

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

$- 5 x^{\frac{2}{3} + 2} = - 1$

Step 4.2.3

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

$- 5 x^{\frac{2}{3} + 2 \cdot \frac{3}{3}} = - 1$

Step 4.2.4

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

$- 5 x^{\frac{2}{3} + \frac{2 \cdot 3}{3}} = - 1$

Step 4.2.5

Combine the numerators over the common denominator.

$- 5 x^{\frac{2 + 2 \cdot 3}{3}} = - 1$

Step 4.2.6

Simplify the numerator.

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Step 4.2.6.1

Multiply $2$ by $3$ .

$- 5 x^{\frac{2 + 6}{3}} = - 1$

Step 4.2.6.2

Add $2$ and $6$ .

$- 5 x^{\frac{8}{3}} = - 1$

Step 5

Raise each side of the equation to the power of $\frac{3}{8}$ to eliminate the fractional exponent on the left side.

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

Step 6

Simplify the left side.

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Step 6.1

Simplify ${(- 5 x^{\frac{8}{3}})}^{\frac{3}{8}}$ .

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Step 6.1.1

Apply the product rule to $- 5 x^{\frac{8}{3}}$ .

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

Step 6.1.2

Multiply the exponents in ${(x^{\frac{8}{3}})}^{\frac{3}{8}}$ .

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Step 6.1.2.1

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

${(- 5)}^{\frac{3}{8}} x^{\frac{8}{3} \cdot \frac{3}{8}} = \pm {(- 1)}^{\frac{3}{8}}$

Step 6.1.2.2

Cancel the common factor of $8$ .

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Step 6.1.2.2.1

Cancel the common factor.

${(- 5)}^{\frac{3}{8}} x^{\frac{8}{3} \cdot \frac{3}{8}} = \pm {(- 1)}^{\frac{3}{8}}$

Step 6.1.2.2.2

Rewrite the expression.

${(- 5)}^{\frac{3}{8}} x^{\frac{1}{3} \cdot 3} = \pm {(- 1)}^{\frac{3}{8}}$

Step 6.1.2.3

Cancel the common factor of $3$ .

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Step 6.1.2.3.1

Cancel the common factor.

${(- 5)}^{\frac{3}{8}} x^{\frac{1}{3} \cdot 3} = \pm {(- 1)}^{\frac{3}{8}}$

Step 6.1.2.3.2

Rewrite the expression.

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

Step 6.1.3

Simplify.

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

Step 6.1.4

Reorder factors in ${(- 5)}^{\frac{3}{8}} x$ .

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

Step 7

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

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Step 7.1

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

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

Step 7.2

Divide each term in $x {(- 5)}^{\frac{3}{8}} = {(- 1)}^{\frac{3}{8}}$ by ${(- 5)}^{\frac{3}{8}}$ and simplify.

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Step 7.2.1

Divide each term in $x {(- 5)}^{\frac{3}{8}} = {(- 1)}^{\frac{3}{8}}$ by ${(- 5)}^{\frac{3}{8}}$ .

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

Step 7.2.2

Simplify the left side.

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Step 7.2.2.1

Cancel the common factor.

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

Step 7.2.2.2

Divide $x$ by $1$ .

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

Step 7.3

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

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

Step 7.4

Multiply $- 1$ by ${(- 1)}^{\frac{3}{8}}$ by adding the exponents.

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Step 7.4.1

Multiply $- 1$ by ${(- 1)}^{\frac{3}{8}}$ .

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Step 7.4.1.1

Raise $- 1$ to the power of $1$ .

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

Step 7.4.1.2

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

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

Step 7.4.2

Write $1$ as a fraction with a common denominator.

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

Step 7.4.3

Combine the numerators over the common denominator.

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

Step 7.4.4

Add $8$ and $3$ .

$x {(- 5)}^{\frac{3}{8}} = {(- 1)}^{\frac{11}{8}}$

Step 7.5

Divide each term in $x {(- 5)}^{\frac{3}{8}} = {(- 1)}^{\frac{11}{8}}$ by ${(- 5)}^{\frac{3}{8}}$ and simplify.

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Step 7.5.1

Divide each term in $x {(- 5)}^{\frac{3}{8}} = {(- 1)}^{\frac{11}{8}}$ by ${(- 5)}^{\frac{3}{8}}$ .

$\frac{x {(- 5)}^{\frac{3}{8}}}{{(- 5)}^{\frac{3}{8}}} = \frac{{(- 1)}^{\frac{11}{8}}}{{(- 5)}^{\frac{3}{8}}}$

Step 7.5.2

Simplify the left side.

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Step 7.5.2.1

Cancel the common factor.

$\frac{x {(- 5)}^{\frac{3}{8}}}{{(- 5)}^{\frac{3}{8}}} = \frac{{(- 1)}^{\frac{11}{8}}}{{(- 5)}^{\frac{3}{8}}}$

Step 7.5.2.2

Divide $x$ by $1$ .

$x = \frac{{(- 1)}^{\frac{11}{8}}}{{(- 5)}^{\frac{3}{8}}}$

Step 7.6

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

$x = \frac{{(- 1)}^{\frac{3}{8}}}{{(- 5)}^{\frac{3}{8}}}, \frac{{(- 1)}^{\frac{11}{8}}}{{(- 5)}^{\frac{3}{8}}}$

Step 8

Exclude the solutions that do not make $3 = 4 - 5 \sqrt[3]{x^{8}}$ true.

$No solution$