Trigonometry Examples

$\frac{y^{\frac{5}{8} \cdot (y^{\frac{3}{8}} - y^{\frac{11}{8}})}}{y^{\frac{1}{3}} (y^{\frac{2}{3}} - y^{\frac{- 1}{3}})}$

Step 1

Convert expressions with fractional exponents to radicals.

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Step 1.1

Apply the rule $x^{\frac{m}{n}} = \sqrt[n]{x^{m}}$ to rewrite the exponentiation as a radical.

$\frac{y^{\frac{5}{8} \cdot (\sqrt[8]{y^{3}} - y^{\frac{11}{8}})}}{y^{\frac{1}{3}} (y^{\frac{2}{3}} - y^{\frac{- 1}{3}})}$

Step 1.2

Apply the rule $x^{\frac{m}{n}} = \sqrt[n]{x^{m}}$ to rewrite the exponentiation as a radical.

$\frac{y^{\frac{5}{8} \cdot (\sqrt[8]{y^{3}} - \sqrt[8]{y^{11}})}}{y^{\frac{1}{3}} (y^{\frac{2}{3}} - y^{\frac{- 1}{3}})}$

Step 1.3

Apply the rule $x^{\frac{m}{n}} = \sqrt[n]{x^{m}}$ to rewrite the exponentiation as a radical.

$\frac{y^{\frac{5}{8} \cdot (\sqrt[8]{y^{3}} - \sqrt[8]{y^{11}})}}{\sqrt[3]{y^{1}} (y^{\frac{2}{3}} - y^{\frac{- 1}{3}})}$

Step 1.4

Apply the rule $x^{\frac{m}{n}} = \sqrt[n]{x^{m}}$ to rewrite the exponentiation as a radical.

$\frac{y^{\frac{5}{8} \cdot (\sqrt[8]{y^{3}} - \sqrt[8]{y^{11}})}}{\sqrt[3]{y^{1}} (\sqrt[3]{y^{2}} - y^{\frac{- 1}{3}})}$

Step 1.5

Apply the rule $x^{\frac{m}{n}} = \sqrt[n]{x^{m}}$ to rewrite the exponentiation as a radical.

$\frac{y^{\frac{5}{8} \cdot (\sqrt[8]{y^{3}} - \sqrt[8]{y^{11}})}}{\sqrt[3]{y^{1}} (\sqrt[3]{y^{2}} - \sqrt[3]{y^{- 1}})}$

Step 1.6

Anything raised to $1$ is the base itself.

$\frac{y^{\frac{5}{8} \cdot (\sqrt[8]{y^{3}} - \sqrt[8]{y^{11}})}}{\sqrt[3]{y} (\sqrt[3]{y^{2}} - \sqrt[3]{y^{- 1}})}$

$\frac{y^{\frac{5}{8} \cdot (\sqrt[8]{y^{3}} - \sqrt[8]{y^{11}})}}{\sqrt[3]{y} (\sqrt[3]{y^{2}} - \sqrt[3]{y^{- 1}})}$

Step 2

Set the denominator in $\frac{y^{\frac{5}{8} \cdot (\sqrt[8]{y^{3}} - \sqrt[8]{y^{11}})}}{\sqrt[3]{y} (\sqrt[3]{y^{2}} - \sqrt[3]{y^{- 1}})}$ equal to $0$ to find where the expression is undefined.

$\sqrt[3]{y} (\sqrt[3]{y^{2}} - \sqrt[3]{y^{- 1}}) = 0$

Step 3

Solve for $y$ .

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Step 3.1

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

${(\sqrt[3]{y} (\sqrt[3]{y^{2}} - \sqrt[3]{y^{- 1}}))}^{3} = 0^{3}$

Step 3.2

Simplify each side of the equation.

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Step 3.2.1

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

${(y^{\frac{1}{3}} (\sqrt[3]{y^{2}} - \sqrt[3]{y^{- 1}}))}^{3} = 0^{3}$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Simplify ${(y^{\frac{1}{3}} (\sqrt[3]{y^{2}} - \sqrt[3]{y^{- 1}}))}^{3}$ .

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Step 3.2.2.1.1

Simplify each term.

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Step 3.2.2.1.1.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

${(y^{\frac{1}{3}} (\sqrt[3]{y^{2}} - \sqrt[3]{\frac{1}{y}}))}^{3} = 0^{3}$

Step 3.2.2.1.1.2

Rewrite $\sqrt[3]{\frac{1}{y}}$ as $\frac{\sqrt[3]{1}}{\sqrt[3]{y}}$ .

${(y^{\frac{1}{3}} (\sqrt[3]{y^{2}} - \frac{\sqrt[3]{1}}{\sqrt[3]{y}}))}^{3} = 0^{3}$

Step 3.2.2.1.1.3

Any root of $1$ is $1$ .

${(y^{\frac{1}{3}} (\sqrt[3]{y^{2}} - \frac{1}{\sqrt[3]{y}}))}^{3} = 0^{3}$

Step 3.2.2.1.1.4

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

${(y^{\frac{1}{3}} (\sqrt[3]{y^{2}} - (\frac{1}{\sqrt[3]{y}} \cdot \frac{{\sqrt[3]{y}}^{2}}{{\sqrt[3]{y}}^{2}})))}^{3} = 0^{3}$

Step 3.2.2.1.1.5

Combine and simplify the denominator.

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Step 3.2.2.1.1.5.1

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

${(y^{\frac{1}{3}} (\sqrt[3]{y^{2}} - \frac{{\sqrt[3]{y}}^{2}}{\sqrt[3]{y} {\sqrt[3]{y}}^{2}}))}^{3} = 0^{3}$

Step 3.2.2.1.1.5.2

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

${(y^{\frac{1}{3}} (\sqrt[3]{y^{2}} - \frac{{\sqrt[3]{y}}^{2}}{{\sqrt[3]{y}}^{1} {\sqrt[3]{y}}^{2}}))}^{3} = 0^{3}$

Step 3.2.2.1.1.5.3

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

${(y^{\frac{1}{3}} (\sqrt[3]{y^{2}} - \frac{{\sqrt[3]{y}}^{2}}{{\sqrt[3]{y}}^{1 + 2}}))}^{3} = 0^{3}$

Step 3.2.2.1.1.5.4

Add $1$ and $2$ .

${(y^{\frac{1}{3}} (\sqrt[3]{y^{2}} - \frac{{\sqrt[3]{y}}^{2}}{{\sqrt[3]{y}}^{3}}))}^{3} = 0^{3}$

Step 3.2.2.1.1.5.5

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

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Step 3.2.2.1.1.5.5.1

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

${(y^{\frac{1}{3}} (\sqrt[3]{y^{2}} - \frac{{\sqrt[3]{y}}^{2}}{{(y^{\frac{1}{3}})}^{3}}))}^{3} = 0^{3}$

Step 3.2.2.1.1.5.5.2

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

${(y^{\frac{1}{3}} (\sqrt[3]{y^{2}} - \frac{{\sqrt[3]{y}}^{2}}{y^{\frac{1}{3} \cdot 3}}))}^{3} = 0^{3}$

Step 3.2.2.1.1.5.5.3

Combine $\frac{1}{3}$ and $3$ .

${(y^{\frac{1}{3}} (\sqrt[3]{y^{2}} - \frac{{\sqrt[3]{y}}^{2}}{y^{\frac{3}{3}}}))}^{3} = 0^{3}$

Step 3.2.2.1.1.5.5.4

Cancel the common factor of $3$ .

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Step 3.2.2.1.1.5.5.4.1

Cancel the common factor.

${(y^{\frac{1}{3}} (\sqrt[3]{y^{2}} - \frac{{\sqrt[3]{y}}^{2}}{y^{\frac{3}{3}}}))}^{3} = 0^{3}$

Step 3.2.2.1.1.5.5.4.2

Rewrite the expression.

${(y^{\frac{1}{3}} (\sqrt[3]{y^{2}} - \frac{{\sqrt[3]{y}}^{2}}{y^{1}}))}^{3} = 0^{3}$

${(y^{\frac{1}{3}} (\sqrt[3]{y^{2}} - \frac{{\sqrt[3]{y}}^{2}}{y^{1}}))}^{3} = 0^{3}$

Step 3.2.2.1.1.5.5.5

Simplify.

${(y^{\frac{1}{3}} (\sqrt[3]{y^{2}} - \frac{{\sqrt[3]{y}}^{2}}{y}))}^{3} = 0^{3}$

${(y^{\frac{1}{3}} (\sqrt[3]{y^{2}} - \frac{{\sqrt[3]{y}}^{2}}{y}))}^{3} = 0^{3}$

${(y^{\frac{1}{3}} (\sqrt[3]{y^{2}} - \frac{{\sqrt[3]{y}}^{2}}{y}))}^{3} = 0^{3}$

Step 3.2.2.1.1.6

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

${(y^{\frac{1}{3}} (\sqrt[3]{y^{2}} - \frac{\sqrt[3]{y^{2}}}{y}))}^{3} = 0^{3}$

${(y^{\frac{1}{3}} (\sqrt[3]{y^{2}} - \frac{\sqrt[3]{y^{2}}}{y}))}^{3} = 0^{3}$

Step 3.2.2.1.2

Apply the distributive property.

${(y^{\frac{1}{3}} \sqrt[3]{y^{2}} + y^{\frac{1}{3}} (- \frac{\sqrt[3]{y^{2}}}{y}))}^{3} = 0^{3}$

Step 3.2.2.1.3

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

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Step 3.2.2.1.3.1

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

${(y^{\frac{1}{3}} y^{\frac{2}{3}} + y^{\frac{1}{3}} (- \frac{\sqrt[3]{y^{2}}}{y}))}^{3} = 0^{3}$

Step 3.2.2.1.3.2

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

${(y^{\frac{1}{3} + \frac{2}{3}} + y^{\frac{1}{3}} (- \frac{\sqrt[3]{y^{2}}}{y}))}^{3} = 0^{3}$

Step 3.2.2.1.3.3

Combine the numerators over the common denominator.

${(y^{\frac{1 + 2}{3}} + y^{\frac{1}{3}} (- \frac{\sqrt[3]{y^{2}}}{y}))}^{3} = 0^{3}$

Step 3.2.2.1.3.4

Add $1$ and $2$ .

${(y^{\frac{3}{3}} + y^{\frac{1}{3}} (- \frac{\sqrt[3]{y^{2}}}{y}))}^{3} = 0^{3}$

Step 3.2.2.1.3.5

Cancel the common factor of $3$ .

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Step 3.2.2.1.3.5.1

Cancel the common factor.

${(y^{\frac{3}{3}} + y^{\frac{1}{3}} (- \frac{\sqrt[3]{y^{2}}}{y}))}^{3} = 0^{3}$

Step 3.2.2.1.3.5.2

Rewrite the expression.

${(y^{1} + y^{\frac{1}{3}} (- \frac{\sqrt[3]{y^{2}}}{y}))}^{3} = 0^{3}$

${(y^{1} + y^{\frac{1}{3}} (- \frac{\sqrt[3]{y^{2}}}{y}))}^{3} = 0^{3}$

${(y^{1} + y^{\frac{1}{3}} (- \frac{\sqrt[3]{y^{2}}}{y}))}^{3} = 0^{3}$

Step 3.2.2.1.4

Rewrite using the commutative property of multiplication.

${(y^{1} - y^{\frac{1}{3}} \frac{\sqrt[3]{y^{2}}}{y})}^{3} = 0^{3}$

Step 3.2.2.1.5

Simplify each term.

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Step 3.2.2.1.5.1

Simplify.

${(y - y^{\frac{1}{3}} \frac{\sqrt[3]{y^{2}}}{y})}^{3} = 0^{3}$

Step 3.2.2.1.5.2

Combine $\frac{\sqrt[3]{y^{2}}}{y}$ and $y^{\frac{1}{3}}$ .

${(y - \frac{\sqrt[3]{y^{2}} y^{\frac{1}{3}}}{y})}^{3} = 0^{3}$

Step 3.2.2.1.5.3

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

${(y - \frac{y^{\frac{2}{3}} y^{\frac{1}{3}}}{y})}^{3} = 0^{3}$

Step 3.2.2.1.5.4

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

${(y - \frac{y^{\frac{2}{3} + \frac{1}{3}}}{y})}^{3} = 0^{3}$

Step 3.2.2.1.5.5

Combine the numerators over the common denominator.

${(y - \frac{y^{\frac{2 + 1}{3}}}{y})}^{3} = 0^{3}$

Step 3.2.2.1.5.6

Add $2$ and $1$ .

${(y - \frac{y^{\frac{3}{3}}}{y})}^{3} = 0^{3}$

Step 3.2.2.1.5.7

Cancel the common factor of $3$ .

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Step 3.2.2.1.5.7.1

Cancel the common factor.

${(y - \frac{y^{\frac{3}{3}}}{y})}^{3} = 0^{3}$

Step 3.2.2.1.5.7.2

Rewrite the expression.

${(y - \frac{y^{1}}{y})}^{3} = 0^{3}$

${(y - \frac{y^{1}}{y})}^{3} = 0^{3}$

Step 3.2.2.1.5.8

Cancel the common factor of $y^{1}$ and $y$ .

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Step 3.2.2.1.5.8.1

Factor $y$ out of $y^{1}$ .

${(y - \frac{y \cdot 1}{y})}^{3} = 0^{3}$

Step 3.2.2.1.5.8.2

Cancel the common factors.

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Step 3.2.2.1.5.8.2.1

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

${(y - \frac{y \cdot 1}{y^{1}})}^{3} = 0^{3}$

Step 3.2.2.1.5.8.2.2

Factor $y$ out of $y^{1}$ .

${(y - \frac{y \cdot 1}{y \cdot 1})}^{3} = 0^{3}$

Step 3.2.2.1.5.8.2.3

Cancel the common factor.

${(y - \frac{y \cdot 1}{y \cdot 1})}^{3} = 0^{3}$

Step 3.2.2.1.5.8.2.4

Rewrite the expression.

${(y - \frac{1}{1})}^{3} = 0^{3}$

Step 3.2.2.1.5.8.2.5

Rewrite the expression.

${(y - 1 \cdot 1)}^{3} = 0^{3}$

${(y - 1 \cdot 1)}^{3} = 0^{3}$

${(y - 1 \cdot 1)}^{3} = 0^{3}$

Step 3.2.2.1.5.9

Multiply $- 1$ by $1$ .

${(y - 1)}^{3} = 0^{3}$

${(y - 1)}^{3} = 0^{3}$

${(y - 1)}^{3} = 0^{3}$

${(y - 1)}^{3} = 0^{3}$

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Raising $0$ to any positive power yields $0$ .

${(y - 1)}^{3} = 0$

${(y - 1)}^{3} = 0$

${(y - 1)}^{3} = 0$

Step 3.3

Solve for $y$ .

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Step 3.3.1

Set the $y - 1$ equal to $0$ .

$y - 1 = 0$

Step 3.3.2

Add $1$ to both sides of the equation.

$y = 1$

$y = 1$

$y = 1$

Step 4

Set the radicand in $\sqrt[8]{y^{3}}$ less than $0$ to find where the expression is undefined.

$y^{3} < 0$

Step 5

Solve for $y$ .

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Step 5.1

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

$\sqrt[3]{y^{3}} < \sqrt[3]{0}$

Step 5.2

Simplify the equation.

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Step 5.2.1

Simplify the left side.

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Step 5.2.1.1

Pull terms out from under the radical.

$y < \sqrt[3]{0}$

$y < \sqrt[3]{0}$

Step 5.2.2

Simplify the right side.

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Step 5.2.2.1

Simplify $\sqrt[3]{0}$ .

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Step 5.2.2.1.1

Rewrite $0$ as $0^{3}$ .

$y < \sqrt[3]{0^{3}}$

Step 5.2.2.1.2

Pull terms out from under the radical.

$y < 0$

$y < 0$

$y < 0$

$y < 0$

$y < 0$

Step 6

Set the radicand in $\sqrt[8]{y^{11}}$ less than $0$ to find where the expression is undefined.

$y^{11} < 0$

Step 7

Solve for $y$ .

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Step 7.1

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

$\sqrt[11]{y^{11}} < \sqrt[11]{0}$

Step 7.2

Simplify the equation.

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Step 7.2.1

Simplify the left side.

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Step 7.2.1.1

Pull terms out from under the radical.

$y < \sqrt[11]{0}$

$y < \sqrt[11]{0}$

Step 7.2.2

Simplify the right side.

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Step 7.2.2.1

Simplify $\sqrt[11]{0}$ .

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Step 7.2.2.1.1

Rewrite $0$ as $0^{11}$ .

$y < \sqrt[11]{0^{11}}$

Step 7.2.2.1.2

Pull terms out from under the radical.

$y < 0$

$y < 0$

$y < 0$

$y < 0$

$y < 0$

Step 8

Set the base in $y^{- 1}$ equal to $0$ to find where the expression is undefined.

$y = 0$

Step 9

The equation is undefined where the denominator equals $0$ , the argument of a square root is less than $0$ , or the argument of a logarithm is less than or equal to $0$ .

$y < 0, y = 1$

$(- \infty, 0) \cup [1, 1]$

Step 10

$[\begin{array}{c} x^{2} & \frac{1}{2} \\ \sqrt{π} & \int x d x \end{array}]$