Finite Math Examples

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

Step 1

Simplify both sides.

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Step 1.1

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

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

Step 1.2

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

$\frac{1}{x^{\frac{1}{3}} + \frac{1}{y}} = \frac{5}{11}$

Step 1.3

Simplify the denominator.

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Step 1.3.1

To write $x^{\frac{1}{3}}$ as a fraction with a common denominator, multiply by $\frac{y}{y}$ .

$\frac{1}{\frac{x^{\frac{1}{3}} y}{y} + \frac{1}{y}} = \frac{5}{11}$

Step 1.3.2

Combine the numerators over the common denominator.

$\frac{1}{\frac{x^{\frac{1}{3}} y + 1}{y}} = \frac{5}{11}$

Step 1.4

Multiply the numerator by the reciprocal of the denominator.

$1 \frac{y}{x^{\frac{1}{3}} y + 1} = \frac{5}{11}$

Step 1.5

Multiply $\frac{y}{x^{\frac{1}{3}} y + 1}$ by $1$ .

$\frac{y}{x^{\frac{1}{3}} y + 1} = \frac{5}{11}$

Step 1.6

Reorder factors in $\frac{y}{x^{\frac{1}{3}} y + 1} = \frac{5}{11}$ .

$\frac{y}{y x^{\frac{1}{3}} + 1} = \frac{5}{11}$

Step 2

Multiply the numerator of the first fraction by the denominator of the second fraction. Set this equal to the product of the denominator of the first fraction and the numerator of the second fraction.

$y \cdot 11 = (y x^{\frac{1}{3}} + 1) \cdot 5$

Step 3

Solve the equation for $x$ .

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Step 3.1

Rewrite the equation as $(y x^{\frac{1}{3}} + 1) \cdot 5 = y \cdot 11$ .

$(y x^{\frac{1}{3}} + 1) \cdot 5 = y \cdot 11$

Step 3.2

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

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Step 3.2.1

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

$\frac{(y x^{\frac{1}{3}} + 1) \cdot 5}{5} = \frac{y \cdot 11}{5}$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor.

$\frac{(y x^{\frac{1}{3}} + 1) \cdot 5}{5} = \frac{y \cdot 11}{5}$

Step 3.2.2.2

Divide $y x^{\frac{1}{3}} + 1$ by $1$ .

$y x^{\frac{1}{3}} + 1 = \frac{y \cdot 11}{5}$

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Move $11$ to the left of $y$ .

$y x^{\frac{1}{3}} + 1 = \frac{11 y}{5}$

Step 3.3

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

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Step 3.3.1

Subtract $1$ from both sides of the equation.

$y x^{\frac{1}{3}} - \frac{11 y}{5} = - 1$

Step 3.3.2

Add $\frac{11 y}{5}$ to both sides of the equation.

$y x^{\frac{1}{3}} = - 1 + \frac{11 y}{5}$

Step 3.4

Raise each side of the equation to the power of $3$ to eliminate the fractional exponent on the left side.

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

Step 3.5

Simplify the exponent.

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Step 3.5.1

Simplify the left side.

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Step 3.5.1.1

Simplify ${(y x^{\frac{1}{3}})}^{3}$ .

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Step 3.5.1.1.1

Apply the product rule to $y x^{\frac{1}{3}}$ .

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

Step 3.5.1.1.2

Multiply the exponents in ${(x^{\frac{1}{3}})}^{3}$ .

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Step 3.5.1.1.2.1

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

$y^{3} x^{\frac{1}{3} \cdot 3} = {(- 1 + \frac{11 y}{5})}^{3}$

Step 3.5.1.1.2.2

Cancel the common factor of $3$ .

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Step 3.5.1.1.2.2.1

Cancel the common factor.

$y^{3} x^{\frac{1}{3} \cdot 3} = {(- 1 + \frac{11 y}{5})}^{3}$

Step 3.5.1.1.2.2.2

Rewrite the expression.

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

Step 3.5.1.1.3

Simplify.

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

Step 3.5.2

Simplify the right side.

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Step 3.5.2.1

Simplify ${(- 1 + \frac{11 y}{5})}^{3}$ .

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Step 3.5.2.1.1

Use the Binomial Theorem.

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

Step 3.5.2.1.2

Simplify each term.

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Step 3.5.2.1.2.1

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

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

Step 3.5.2.1.2.2

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

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

Step 3.5.2.1.2.3

Multiply $3$ by $1$ .

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

Step 3.5.2.1.2.4

Multiply $3 \frac{11 y}{5}$ .

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Step 3.5.2.1.2.4.1

Combine $3$ and $\frac{11 y}{5}$ .

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

Step 3.5.2.1.2.4.2

Multiply $11$ by $3$ .

$y^{3} x = - 1 + \frac{33 y}{5} + 3 \cdot - 1 {(\frac{11 y}{5})}^{2} + {(\frac{11 y}{5})}^{3}$

Step 3.5.2.1.2.5

Multiply $3$ by $- 1$ .

$y^{3} x = - 1 + \frac{33 y}{5} - 3 {(\frac{11 y}{5})}^{2} + {(\frac{11 y}{5})}^{3}$

Step 3.5.2.1.2.6

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

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Step 3.5.2.1.2.6.1

Apply the product rule to $\frac{11 y}{5}$ .

$y^{3} x = - 1 + \frac{33 y}{5} - 3 \frac{{(11 y)}^{2}}{5^{2}} + {(\frac{11 y}{5})}^{3}$

Step 3.5.2.1.2.6.2

Apply the product rule to $11 y$ .

$y^{3} x = - 1 + \frac{33 y}{5} - 3 \frac{11^{2} y^{2}}{5^{2}} + {(\frac{11 y}{5})}^{3}$

Step 3.5.2.1.2.7

Raise $11$ to the power of $2$ .

$y^{3} x = - 1 + \frac{33 y}{5} - 3 \frac{121 y^{2}}{5^{2}} + {(\frac{11 y}{5})}^{3}$

Step 3.5.2.1.2.8

Raise $5$ to the power of $2$ .

$y^{3} x = - 1 + \frac{33 y}{5} - 3 \frac{121 y^{2}}{25} + {(\frac{11 y}{5})}^{3}$

Step 3.5.2.1.2.9

Multiply $- 3 \frac{121 y^{2}}{25}$ .

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Step 3.5.2.1.2.9.1

Combine $- 3$ and $\frac{121 y^{2}}{25}$ .

$y^{3} x = - 1 + \frac{33 y}{5} + \frac{- 3 (121 y^{2})}{25} + {(\frac{11 y}{5})}^{3}$

Step 3.5.2.1.2.9.2

Multiply $121$ by $- 3$ .

$y^{3} x = - 1 + \frac{33 y}{5} + \frac{- 363 y^{2}}{25} + {(\frac{11 y}{5})}^{3}$

Step 3.5.2.1.2.10

Move the negative in front of the fraction.

$y^{3} x = - 1 + \frac{33 y}{5} - \frac{363 y^{2}}{25} + {(\frac{11 y}{5})}^{3}$

Step 3.5.2.1.2.11

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

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Step 3.5.2.1.2.11.1

Apply the product rule to $\frac{11 y}{5}$ .

$y^{3} x = - 1 + \frac{33 y}{5} - \frac{363 y^{2}}{25} + \frac{{(11 y)}^{3}}{5^{3}}$

Step 3.5.2.1.2.11.2

Apply the product rule to $11 y$ .

$y^{3} x = - 1 + \frac{33 y}{5} - \frac{363 y^{2}}{25} + \frac{11^{3} y^{3}}{5^{3}}$

Step 3.5.2.1.2.12

Raise $11$ to the power of $3$ .

$y^{3} x = - 1 + \frac{33 y}{5} - \frac{363 y^{2}}{25} + \frac{1331 y^{3}}{5^{3}}$

Step 3.5.2.1.2.13

Raise $5$ to the power of $3$ .

$y^{3} x = - 1 + \frac{33 y}{5} - \frac{363 y^{2}}{25} + \frac{1331 y^{3}}{125}$

Step 3.6

Divide each term in $y^{3} x = - 1 + \frac{33 y}{5} - \frac{363 y^{2}}{25} + \frac{1331 y^{3}}{125}$ by $y^{3}$ and simplify.

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Step 3.6.1

Divide each term in $y^{3} x = - 1 + \frac{33 y}{5} - \frac{363 y^{2}}{25} + \frac{1331 y^{3}}{125}$ by $y^{3}$ .

$\frac{y^{3} x}{y^{3}} = \frac{- 1}{y^{3}} + \frac{\frac{33 y}{5}}{y^{3}} + \frac{- \frac{363 y^{2}}{25}}{y^{3}} + \frac{\frac{1331 y^{3}}{125}}{y^{3}}$

Step 3.6.2

Simplify the left side.

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Step 3.6.2.1

Cancel the common factor of $y^{3}$ .

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Step 3.6.2.1.1

Cancel the common factor.

$\frac{y^{3} x}{y^{3}} = \frac{- 1}{y^{3}} + \frac{\frac{33 y}{5}}{y^{3}} + \frac{- \frac{363 y^{2}}{25}}{y^{3}} + \frac{\frac{1331 y^{3}}{125}}{y^{3}}$

Step 3.6.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 1}{y^{3}} + \frac{\frac{33 y}{5}}{y^{3}} + \frac{- \frac{363 y^{2}}{25}}{y^{3}} + \frac{\frac{1331 y^{3}}{125}}{y^{3}}$

Step 3.6.3

Simplify the right side.

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Step 3.6.3.1

Simplify each term.

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Step 3.6.3.1.1

Move the negative in front of the fraction.

$x = - \frac{1}{y^{3}} + \frac{\frac{33 y}{5}}{y^{3}} + \frac{- \frac{363 y^{2}}{25}}{y^{3}} + \frac{\frac{1331 y^{3}}{125}}{y^{3}}$

Step 3.6.3.1.2

Multiply the numerator by the reciprocal of the denominator.

$x = - \frac{1}{y^{3}} + \frac{33 y}{5} \cdot \frac{1}{y^{3}} + \frac{- \frac{363 y^{2}}{25}}{y^{3}} + \frac{\frac{1331 y^{3}}{125}}{y^{3}}$

Step 3.6.3.1.3

Combine.

$x = - \frac{1}{y^{3}} + \frac{33 y \cdot 1}{5 y^{3}} + \frac{- \frac{363 y^{2}}{25}}{y^{3}} + \frac{\frac{1331 y^{3}}{125}}{y^{3}}$

Step 3.6.3.1.4

Cancel the common factor of $y$ and $y^{3}$ .

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Step 3.6.3.1.4.1

Factor $y$ out of $33 y \cdot 1$ .

$x = - \frac{1}{y^{3}} + \frac{y (33 \cdot 1)}{5 y^{3}} + \frac{- \frac{363 y^{2}}{25}}{y^{3}} + \frac{\frac{1331 y^{3}}{125}}{y^{3}}$

Step 3.6.3.1.4.2

Cancel the common factors.

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Step 3.6.3.1.4.2.1

Factor $y$ out of $5 y^{3}$ .

$x = - \frac{1}{y^{3}} + \frac{y (33 \cdot 1)}{y (5 y^{2})} + \frac{- \frac{363 y^{2}}{25}}{y^{3}} + \frac{\frac{1331 y^{3}}{125}}{y^{3}}$

Step 3.6.3.1.4.2.2

Cancel the common factor.

$x = - \frac{1}{y^{3}} + \frac{y (33 \cdot 1)}{y (5 y^{2})} + \frac{- \frac{363 y^{2}}{25}}{y^{3}} + \frac{\frac{1331 y^{3}}{125}}{y^{3}}$

Step 3.6.3.1.4.2.3

Rewrite the expression.

$x = - \frac{1}{y^{3}} + \frac{33 \cdot 1}{5 y^{2}} + \frac{- \frac{363 y^{2}}{25}}{y^{3}} + \frac{\frac{1331 y^{3}}{125}}{y^{3}}$

Step 3.6.3.1.5

Multiply $33$ by $1$ .

$x = - \frac{1}{y^{3}} + \frac{33}{5 y^{2}} + \frac{- \frac{363 y^{2}}{25}}{y^{3}} + \frac{\frac{1331 y^{3}}{125}}{y^{3}}$

Step 3.6.3.1.6

Multiply the numerator by the reciprocal of the denominator.

$x = - \frac{1}{y^{3}} + \frac{33}{5 y^{2}} - \frac{363 y^{2}}{25} \cdot \frac{1}{y^{3}} + \frac{\frac{1331 y^{3}}{125}}{y^{3}}$

Step 3.6.3.1.7

Cancel the common factor of $y^{2}$ .

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Step 3.6.3.1.7.1

Move the leading negative in $- \frac{363 y^{2}}{25}$ into the numerator.

$x = - \frac{1}{y^{3}} + \frac{33}{5 y^{2}} + \frac{- 363 y^{2}}{25} \cdot \frac{1}{y^{3}} + \frac{\frac{1331 y^{3}}{125}}{y^{3}}$

Step 3.6.3.1.7.2

Factor $y^{2}$ out of $- 363 y^{2}$ .

$x = - \frac{1}{y^{3}} + \frac{33}{5 y^{2}} + \frac{y^{2} \cdot - 363}{25} \cdot \frac{1}{y^{3}} + \frac{\frac{1331 y^{3}}{125}}{y^{3}}$

Step 3.6.3.1.7.3

Factor $y^{2}$ out of $y^{3}$ .

$x = - \frac{1}{y^{3}} + \frac{33}{5 y^{2}} + \frac{y^{2} \cdot - 363}{25} \cdot \frac{1}{y^{2} y} + \frac{\frac{1331 y^{3}}{125}}{y^{3}}$

Step 3.6.3.1.7.4

Cancel the common factor.

$x = - \frac{1}{y^{3}} + \frac{33}{5 y^{2}} + \frac{y^{2} \cdot - 363}{25} \cdot \frac{1}{y^{2} y} + \frac{\frac{1331 y^{3}}{125}}{y^{3}}$

Step 3.6.3.1.7.5

Rewrite the expression.

$x = - \frac{1}{y^{3}} + \frac{33}{5 y^{2}} + \frac{- 363}{25} \cdot \frac{1}{y} + \frac{\frac{1331 y^{3}}{125}}{y^{3}}$

Step 3.6.3.1.8

Multiply $\frac{- 363}{25}$ by $\frac{1}{y}$ .

$x = - \frac{1}{y^{3}} + \frac{33}{5 y^{2}} + \frac{- 363}{25 y} + \frac{\frac{1331 y^{3}}{125}}{y^{3}}$

Step 3.6.3.1.9

Move the negative in front of the fraction.

$x = - \frac{1}{y^{3}} + \frac{33}{5 y^{2}} - \frac{363}{25 y} + \frac{\frac{1331 y^{3}}{125}}{y^{3}}$

Step 3.6.3.1.10

Multiply the numerator by the reciprocal of the denominator.

$x = - \frac{1}{y^{3}} + \frac{33}{5 y^{2}} - \frac{363}{25 y} + \frac{1331 y^{3}}{125} \cdot \frac{1}{y^{3}}$

Step 3.6.3.1.11

Combine.

$x = - \frac{1}{y^{3}} + \frac{33}{5 y^{2}} - \frac{363}{25 y} + \frac{1331 y^{3} \cdot 1}{125 y^{3}}$

Step 3.6.3.1.12

Cancel the common factor of $y^{3}$ .

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Step 3.6.3.1.12.1

Cancel the common factor.

$x = - \frac{1}{y^{3}} + \frac{33}{5 y^{2}} - \frac{363}{25 y} + \frac{1331 y^{3} \cdot 1}{125 y^{3}}$

Step 3.6.3.1.12.2

Rewrite the expression.

$x = - \frac{1}{y^{3}} + \frac{33}{5 y^{2}} - \frac{363}{25 y} + \frac{1331 \cdot 1}{125}$

Step 3.6.3.1.13

Multiply $1331$ by $1$ .

$x = - \frac{1}{y^{3}} + \frac{33}{5 y^{2}} - \frac{363}{25 y} + \frac{1331}{125}$