Algebra Examples

$y = a \sqrt[3]{x - h} + k$

Step 1

Rewrite the equation as $a \sqrt[3]{x - h} + k = y$ .

$a \sqrt[3]{x - h} + k = y$

Step 2

Subtract $k$ from both sides of the equation.

$a \sqrt[3]{x - h} = y - k$

Step 3

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

${(a \sqrt[3]{x - h})}^{3} = {(y - k)}^{3}$

Step 4

Simplify each side of the equation.

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Step 4.1

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

${(a {(x - h)}^{\frac{1}{3}})}^{3} = {(y - k)}^{3}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Simplify ${(a {(x - h)}^{\frac{1}{3}})}^{3}$ .

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Step 4.2.1.1

Apply the product rule to $a {(x - h)}^{\frac{1}{3}}$ .

$a^{3} {({(x - h)}^{\frac{1}{3}})}^{3} = {(y - k)}^{3}$

Step 4.2.1.2

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

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Step 4.2.1.2.1

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

$a^{3} {(x - h)}^{\frac{1}{3} \cdot 3} = {(y - k)}^{3}$

Step 4.2.1.2.2

Cancel the common factor of $3$ .

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Step 4.2.1.2.2.1

Cancel the common factor.

$a^{3} {(x - h)}^{\frac{1}{3} \cdot 3} = {(y - k)}^{3}$

Step 4.2.1.2.2.2

Rewrite the expression.

$a^{3} {(x - h)}^{1} = {(y - k)}^{3}$

Step 4.2.1.3

Simplify.

$a^{3} (x - h) = {(y - k)}^{3}$

Step 4.2.1.4

Apply the distributive property.

$a^{3} x + a^{3} (- h) = {(y - k)}^{3}$

Step 4.2.1.5

Rewrite using the commutative property of multiplication.

$a^{3} x - a^{3} h = {(y - k)}^{3}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Simplify ${(y - k)}^{3}$ .

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Step 4.3.1.1

Use the Binomial Theorem.

$a^{3} x - a^{3} h = y^{3} + 3 y^{2} (- k) + 3 y {(- k)}^{2} + {(- k)}^{3}$

Step 4.3.1.2

Simplify each term.

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Step 4.3.1.2.1

Rewrite using the commutative property of multiplication.

$a^{3} x - a^{3} h = y^{3} + 3 \cdot - 1 y^{2} k + 3 y {(- k)}^{2} + {(- k)}^{3}$

Step 4.3.1.2.2

Multiply $3$ by $- 1$ .

$a^{3} x - a^{3} h = y^{3} - 3 y^{2} k + 3 y {(- k)}^{2} + {(- k)}^{3}$

Step 4.3.1.2.3

Apply the product rule to $- k$ .

$a^{3} x - a^{3} h = y^{3} - 3 y^{2} k + 3 y ({(- 1)}^{2} k^{2}) + {(- k)}^{3}$

Step 4.3.1.2.4

Rewrite using the commutative property of multiplication.

$a^{3} x - a^{3} h = y^{3} - 3 y^{2} k + 3 \cdot {(- 1)}^{2} y k^{2} + {(- k)}^{3}$

Step 4.3.1.2.5

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

$a^{3} x - a^{3} h = y^{3} - 3 y^{2} k + 3 \cdot 1 y k^{2} + {(- k)}^{3}$

Step 4.3.1.2.6

Multiply $3$ by $1$ .

$a^{3} x - a^{3} h = y^{3} - 3 y^{2} k + 3 y k^{2} + {(- k)}^{3}$

Step 4.3.1.2.7

Apply the product rule to $- k$ .

$a^{3} x - a^{3} h = y^{3} - 3 y^{2} k + 3 y k^{2} + {(- 1)}^{3} k^{3}$

Step 4.3.1.2.8

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

$a^{3} x - a^{3} h = y^{3} - 3 y^{2} k + 3 y k^{2} - k^{3}$

Step 5

Solve for $h$ .

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Step 5.1

Subtract $a^{3} x$ from both sides of the equation.

$- a^{3} h = y^{3} - 3 y^{2} k + 3 y k^{2} - k^{3} - a^{3} x$

Step 5.2

Divide each term in $- a^{3} h = y^{3} - 3 y^{2} k + 3 y k^{2} - k^{3} - a^{3} x$ by $- a^{3}$ and simplify.

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Step 5.2.1

Divide each term in $- a^{3} h = y^{3} - 3 y^{2} k + 3 y k^{2} - k^{3} - a^{3} x$ by $- a^{3}$ .

$\frac{- a^{3} h}{- a^{3}} = \frac{y^{3}}{- a^{3}} + \frac{- 3 y^{2} k}{- a^{3}} + \frac{3 y k^{2}}{- a^{3}} + \frac{- k^{3}}{- a^{3}} + \frac{- a^{3} x}{- a^{3}}$

Step 5.2.2

Simplify the left side.

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Step 5.2.2.1

Dividing two negative values results in a positive value.

$\frac{a^{3} h}{a^{3}} = \frac{y^{3}}{- a^{3}} + \frac{- 3 y^{2} k}{- a^{3}} + \frac{3 y k^{2}}{- a^{3}} + \frac{- k^{3}}{- a^{3}} + \frac{- a^{3} x}{- a^{3}}$

Step 5.2.2.2

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

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Step 5.2.2.2.1

Cancel the common factor.

$\frac{a^{3} h}{a^{3}} = \frac{y^{3}}{- a^{3}} + \frac{- 3 y^{2} k}{- a^{3}} + \frac{3 y k^{2}}{- a^{3}} + \frac{- k^{3}}{- a^{3}} + \frac{- a^{3} x}{- a^{3}}$

Step 5.2.2.2.2

Divide $h$ by $1$ .

$h = \frac{y^{3}}{- a^{3}} + \frac{- 3 y^{2} k}{- a^{3}} + \frac{3 y k^{2}}{- a^{3}} + \frac{- k^{3}}{- a^{3}} + \frac{- a^{3} x}{- a^{3}}$

Step 5.2.3

Simplify the right side.

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Step 5.2.3.1

Simplify each term.

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Step 5.2.3.1.1

Move the negative in front of the fraction.

$h = - \frac{y^{3}}{a^{3}} + \frac{- 3 y^{2} k}{- a^{3}} + \frac{3 y k^{2}}{- a^{3}} + \frac{- k^{3}}{- a^{3}} + \frac{- a^{3} x}{- a^{3}}$

Step 5.2.3.1.2

Dividing two negative values results in a positive value.

$h = - \frac{y^{3}}{a^{3}} + \frac{3 y^{2} k}{a^{3}} + \frac{3 y k^{2}}{- a^{3}} + \frac{- k^{3}}{- a^{3}} + \frac{- a^{3} x}{- a^{3}}$

Step 5.2.3.1.3

Move the negative in front of the fraction.

$h = - \frac{y^{3}}{a^{3}} + \frac{3 y^{2} k}{a^{3}} - \frac{3 y k^{2}}{a^{3}} + \frac{- k^{3}}{- a^{3}} + \frac{- a^{3} x}{- a^{3}}$

Step 5.2.3.1.4

Dividing two negative values results in a positive value.

$h = - \frac{y^{3}}{a^{3}} + \frac{3 y^{2} k}{a^{3}} - \frac{3 y k^{2}}{a^{3}} + \frac{k^{3}}{a^{3}} + \frac{- a^{3} x}{- a^{3}}$

Step 5.2.3.1.5

Dividing two negative values results in a positive value.

$h = - \frac{y^{3}}{a^{3}} + \frac{3 y^{2} k}{a^{3}} - \frac{3 y k^{2}}{a^{3}} + \frac{k^{3}}{a^{3}} + \frac{a^{3} x}{a^{3}}$

Step 5.2.3.1.6

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

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Step 5.2.3.1.6.1

Cancel the common factor.

$h = - \frac{y^{3}}{a^{3}} + \frac{3 y^{2} k}{a^{3}} - \frac{3 y k^{2}}{a^{3}} + \frac{k^{3}}{a^{3}} + \frac{a^{3} x}{a^{3}}$

Step 5.2.3.1.6.2

Divide $x$ by $1$ .

$h = - \frac{y^{3}}{a^{3}} + \frac{3 y^{2} k}{a^{3}} - \frac{3 y k^{2}}{a^{3}} + \frac{k^{3}}{a^{3}} + x$