Finite Math Examples

$\sqrt[3]{x^{2}} + \sqrt[3]{y^{2}} = 4$

Step 1

Subtract $\sqrt[3]{y^{2}}$ from both sides of the equation.

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

Step 2

To remove the radical on the left side of the equation, cube both sides of the equation.

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

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 $\sqrt[3]{x^{2}}$ as $x^{\frac{2}{3}}$ .

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

Step 3.2

Simplify the left side.

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Step 3.2.1

Multiply the exponents in ${(x^{\frac{2}{3}})}^{3}$ .

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Step 3.2.1.1

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

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

Step 3.2.1.2

Cancel the common factor of $3$ .

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Step 3.2.1.2.1

Cancel the common factor.

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

Step 3.2.1.2.2

Rewrite the expression.

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

Step 3.3

Simplify the right side.

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Step 3.3.1

Simplify ${(4 - \sqrt[3]{y^{2}})}^{3}$ .

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Step 3.3.1.1

Use the Binomial Theorem.

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

Step 3.3.1.2

Simplify each term.

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Step 3.3.1.2.1

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

$x^{2} = 64 + 3 \cdot 4^{2} (- \sqrt[3]{y^{2}}) + 3 \cdot 4 {(- \sqrt[3]{y^{2}})}^{2} + {(- \sqrt[3]{y^{2}})}^{3}$

Step 3.3.1.2.2

Raise $4$ to the power of $2$ .

$x^{2} = 64 + 3 \cdot 16 (- \sqrt[3]{y^{2}}) + 3 \cdot 4 {(- \sqrt[3]{y^{2}})}^{2} + {(- \sqrt[3]{y^{2}})}^{3}$

Step 3.3.1.2.3

Multiply $3$ by $16$ .

$x^{2} = 64 + 48 (- \sqrt[3]{y^{2}}) + 3 \cdot 4 {(- \sqrt[3]{y^{2}})}^{2} + {(- \sqrt[3]{y^{2}})}^{3}$

Step 3.3.1.2.4

Multiply $- 1$ by $48$ .

$x^{2} = 64 - 48 \sqrt[3]{y^{2}} + 3 \cdot 4 {(- \sqrt[3]{y^{2}})}^{2} + {(- \sqrt[3]{y^{2}})}^{3}$

Step 3.3.1.2.5

Multiply $3$ by $4$ .

$x^{2} = 64 - 48 \sqrt[3]{y^{2}} + 12 {(- \sqrt[3]{y^{2}})}^{2} + {(- \sqrt[3]{y^{2}})}^{3}$

Step 3.3.1.2.6

Apply the product rule to $- \sqrt[3]{y^{2}}$ .

$x^{2} = 64 - 48 \sqrt[3]{y^{2}} + 12 ({(- 1)}^{2} {\sqrt[3]{y^{2}}}^{2}) + {(- \sqrt[3]{y^{2}})}^{3}$

Step 3.3.1.2.7

Raise $- 1$ to the power of $2$ .

$x^{2} = 64 - 48 \sqrt[3]{y^{2}} + 12 (1 {\sqrt[3]{y^{2}}}^{2}) + {(- \sqrt[3]{y^{2}})}^{3}$

Step 3.3.1.2.8

Multiply ${\sqrt[3]{y^{2}}}^{2}$ by $1$ .

$x^{2} = 64 - 48 \sqrt[3]{y^{2}} + 12 {\sqrt[3]{y^{2}}}^{2} + {(- \sqrt[3]{y^{2}})}^{3}$

Step 3.3.1.2.9

Rewrite ${\sqrt[3]{y^{2}}}^{2}$ as $\sqrt[3]{{(y^{2})}^{2}}$ .

$x^{2} = 64 - 48 \sqrt[3]{y^{2}} + 12 \sqrt[3]{{(y^{2})}^{2}} + {(- \sqrt[3]{y^{2}})}^{3}$

Step 3.3.1.2.10

Multiply the exponents in ${(y^{2})}^{2}$ .

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Step 3.3.1.2.10.1

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

$x^{2} = 64 - 48 \sqrt[3]{y^{2}} + 12 \sqrt[3]{y^{2 \cdot 2}} + {(- \sqrt[3]{y^{2}})}^{3}$

Step 3.3.1.2.10.2

Multiply $2$ by $2$ .

$x^{2} = 64 - 48 \sqrt[3]{y^{2}} + 12 \sqrt[3]{y^{4}} + {(- \sqrt[3]{y^{2}})}^{3}$

Step 3.3.1.2.11

Factor out $y^{3}$ .

$x^{2} = 64 - 48 \sqrt[3]{y^{2}} + 12 \sqrt[3]{y^{3} y} + {(- \sqrt[3]{y^{2}})}^{3}$

Step 3.3.1.2.12

Pull terms out from under the radical.

$x^{2} = 64 - 48 \sqrt[3]{y^{2}} + 12 (y \sqrt[3]{y}) + {(- \sqrt[3]{y^{2}})}^{3}$

Step 3.3.1.2.13

Apply the product rule to $- \sqrt[3]{y^{2}}$ .

$x^{2} = 64 - 48 \sqrt[3]{y^{2}} + 12 y \sqrt[3]{y} + {(- 1)}^{3} {\sqrt[3]{y^{2}}}^{3}$

Step 3.3.1.2.14

Raise $- 1$ to the power of $3$ .

$x^{2} = 64 - 48 \sqrt[3]{y^{2}} + 12 y \sqrt[3]{y} - {\sqrt[3]{y^{2}}}^{3}$

Step 3.3.1.2.15

Rewrite ${\sqrt[3]{y^{2}}}^{3}$ as $y^{2}$ .

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Step 3.3.1.2.15.1

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

$x^{2} = 64 - 48 \sqrt[3]{y^{2}} + 12 y \sqrt[3]{y} - {(y^{\frac{2}{3}})}^{3}$

Step 3.3.1.2.15.2

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

$x^{2} = 64 - 48 \sqrt[3]{y^{2}} + 12 y \sqrt[3]{y} - y^{\frac{2}{3} \cdot 3}$

Step 3.3.1.2.15.3

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

$x^{2} = 64 - 48 \sqrt[3]{y^{2}} + 12 y \sqrt[3]{y} - y^{\frac{2 \cdot 3}{3}}$

Step 3.3.1.2.15.4

Multiply $2$ by $3$ .

$x^{2} = 64 - 48 \sqrt[3]{y^{2}} + 12 y \sqrt[3]{y} - y^{\frac{6}{3}}$

Step 3.3.1.2.15.5

Cancel the common factor of $6$ and $3$ .

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Step 3.3.1.2.15.5.1

Factor $3$ out of $6$ .

$x^{2} = 64 - 48 \sqrt[3]{y^{2}} + 12 y \sqrt[3]{y} - y^{\frac{3 \cdot 2}{3}}$

Step 3.3.1.2.15.5.2

Cancel the common factors.

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Step 3.3.1.2.15.5.2.1

Factor $3$ out of $3$ .

$x^{2} = 64 - 48 \sqrt[3]{y^{2}} + 12 y \sqrt[3]{y} - y^{\frac{3 \cdot 2}{3 (1)}}$

Step 3.3.1.2.15.5.2.2

Cancel the common factor.

$x^{2} = 64 - 48 \sqrt[3]{y^{2}} + 12 y \sqrt[3]{y} - y^{\frac{3 \cdot 2}{3 \cdot 1}}$

Step 3.3.1.2.15.5.2.3

Rewrite the expression.

$x^{2} = 64 - 48 \sqrt[3]{y^{2}} + 12 y \sqrt[3]{y} - y^{\frac{2}{1}}$

Step 3.3.1.2.15.5.2.4

Divide $2$ by $1$ .

$x^{2} = 64 - 48 \sqrt[3]{y^{2}} + 12 y \sqrt[3]{y} - y^{2}$

Step 4

Solve for $x$ .

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Step 4.1

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

$x = \pm \sqrt{64 - 48 \sqrt[3]{y^{2}} + 12 y \sqrt[3]{y} - y^{2}}$

Step 4.2

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

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Step 4.2.1

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

$x = \sqrt{64 - 48 \sqrt[3]{y^{2}} + 12 y \sqrt[3]{y} - y^{2}}$

Step 4.2.2

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

$x = - \sqrt{64 - 48 \sqrt[3]{y^{2}} + 12 y \sqrt[3]{y} - y^{2}}$

Step 4.2.3

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

$x = \sqrt{64 - 48 \sqrt[3]{y^{2}} + 12 y \sqrt[3]{y} - y^{2}}$

$x = - \sqrt{64 - 48 \sqrt[3]{y^{2}} + 12 y \sqrt[3]{y} - y^{2}}$

$x = \sqrt{64 - 48 \sqrt[3]{y^{2}} + 12 y \sqrt[3]{y} - y^{2}}$

$x = - \sqrt{64 - 48 \sqrt[3]{y^{2}} + 12 y \sqrt[3]{y} - y^{2}}$

$x = \sqrt{64 - 48 \sqrt[3]{y^{2}} + 12 y \sqrt[3]{y} - y^{2}}$

$x = - \sqrt{64 - 48 \sqrt[3]{y^{2}} + 12 y \sqrt[3]{y} - y^{2}}$