Finite Math Examples

$y = 6 {(4 x + 9)}^{8}$

Step 1

Rewrite the equation as $6 {(4 x + 9)}^{8} = y$ .

$6 {(4 x + 9)}^{8} = y$

Step 2

Divide each term in $6 {(4 x + 9)}^{8} = y$ by $6$ and simplify.

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Step 2.1

Divide each term in $6 {(4 x + 9)}^{8} = y$ by $6$ .

$\frac{6 {(4 x + 9)}^{8}}{6} = \frac{y}{6}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $6$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{6 {(4 x + 9)}^{8}}{6} = \frac{y}{6}$

Step 2.2.1.2

Divide ${(4 x + 9)}^{8}$ by $1$ .

${(4 x + 9)}^{8} = \frac{y}{6}$

Step 3

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

$4 x + 9 = \pm \sqrt[8]{\frac{y}{6}}$

Step 4

Rewrite $\sqrt[8]{\frac{y}{6}}$ as $\frac{\sqrt[8]{y}}{\sqrt[8]{6}}$ .

$4 x + 9 = \pm \frac{\sqrt[8]{y}}{\sqrt[8]{6}}$

Step 5

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

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Step 5.1

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

$4 x + 9 = \frac{\sqrt[8]{y}}{\sqrt[8]{6}}$

Step 5.2

Subtract $9$ from both sides of the equation.

$4 x = \frac{\sqrt[8]{y}}{\sqrt[8]{6}} - 9$

Step 5.3

Divide each term in $4 x = \frac{\sqrt[8]{y}}{\sqrt[8]{6}} - 9$ by $4$ and simplify.

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Step 5.3.1

Divide each term in $4 x = \frac{\sqrt[8]{y}}{\sqrt[8]{6}} - 9$ by $4$ .

$\frac{4 x}{4} = \frac{\frac{\sqrt[8]{y}}{\sqrt[8]{6}}}{4} + \frac{- 9}{4}$

Step 5.3.2

Simplify the left side.

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Step 5.3.2.1

Cancel the common factor of $4$ .

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Step 5.3.2.1.1

Cancel the common factor.

$\frac{4 x}{4} = \frac{\frac{\sqrt[8]{y}}{\sqrt[8]{6}}}{4} + \frac{- 9}{4}$

Step 5.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{\frac{\sqrt[8]{y}}{\sqrt[8]{6}}}{4} + \frac{- 9}{4}$

Step 5.3.3

Simplify the right side.

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Step 5.3.3.1

Simplify each term.

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Step 5.3.3.1.1

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{\sqrt[8]{y}}{\sqrt[8]{6}} \cdot \frac{1}{4} + \frac{- 9}{4}$

Step 5.3.3.1.2

Multiply $\frac{\sqrt[8]{y}}{\sqrt[8]{6}}$ by $\frac{1}{4}$ .

$x = \frac{\sqrt[8]{y}}{\sqrt[8]{6} \cdot 4} + \frac{- 9}{4}$

Step 5.3.3.1.3

Move $4$ to the left of $\sqrt[8]{6}$ .

$x = \frac{\sqrt[8]{y}}{4 \sqrt[8]{6}} + \frac{- 9}{4}$

Step 5.3.3.1.4

Multiply $\frac{\sqrt[8]{y}}{4 \sqrt[8]{6}}$ by $\frac{{\sqrt[8]{6}}^{7}}{{\sqrt[8]{6}}^{7}}$ .

$x = \frac{\sqrt[8]{y}}{4 \sqrt[8]{6}} \cdot \frac{{\sqrt[8]{6}}^{7}}{{\sqrt[8]{6}}^{7}} + \frac{- 9}{4}$

Step 5.3.3.1.5

Combine and simplify the denominator.

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Step 5.3.3.1.5.1

Multiply $\frac{\sqrt[8]{y}}{4 \sqrt[8]{6}}$ by $\frac{{\sqrt[8]{6}}^{7}}{{\sqrt[8]{6}}^{7}}$ .

$x = \frac{\sqrt[8]{y} {\sqrt[8]{6}}^{7}}{4 \sqrt[8]{6} {\sqrt[8]{6}}^{7}} + \frac{- 9}{4}$

Step 5.3.3.1.5.2

Move $\sqrt[8]{6}$ .

$x = \frac{\sqrt[8]{y} {\sqrt[8]{6}}^{7}}{4 (\sqrt[8]{6} {\sqrt[8]{6}}^{7})} + \frac{- 9}{4}$

Step 5.3.3.1.5.3

Raise $\sqrt[8]{6}$ to the power of $1$ .

$x = \frac{\sqrt[8]{y} {\sqrt[8]{6}}^{7}}{4 ({\sqrt[8]{6}}^{1} {\sqrt[8]{6}}^{7})} + \frac{- 9}{4}$

Step 5.3.3.1.5.4

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

$x = \frac{\sqrt[8]{y} {\sqrt[8]{6}}^{7}}{4 {\sqrt[8]{6}}^{1 + 7}} + \frac{- 9}{4}$

Step 5.3.3.1.5.5

Add $1$ and $7$ .

$x = \frac{\sqrt[8]{y} {\sqrt[8]{6}}^{7}}{4 {\sqrt[8]{6}}^{8}} + \frac{- 9}{4}$

Step 5.3.3.1.5.6

Rewrite ${\sqrt[8]{6}}^{8}$ as $6$ .

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Step 5.3.3.1.5.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[8]{6}$ as $6^{\frac{1}{8}}$ .

$x = \frac{\sqrt[8]{y} {\sqrt[8]{6}}^{7}}{4 {(6^{\frac{1}{8}})}^{8}} + \frac{- 9}{4}$

Step 5.3.3.1.5.6.2

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

$x = \frac{\sqrt[8]{y} {\sqrt[8]{6}}^{7}}{4 \cdot 6^{\frac{1}{8} \cdot 8}} + \frac{- 9}{4}$

Step 5.3.3.1.5.6.3

Combine $\frac{1}{8}$ and $8$ .

$x = \frac{\sqrt[8]{y} {\sqrt[8]{6}}^{7}}{4 \cdot 6^{\frac{8}{8}}} + \frac{- 9}{4}$

Step 5.3.3.1.5.6.4

Cancel the common factor of $8$ .

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Step 5.3.3.1.5.6.4.1

Cancel the common factor.

$x = \frac{\sqrt[8]{y} {\sqrt[8]{6}}^{7}}{4 \cdot 6^{\frac{8}{8}}} + \frac{- 9}{4}$

Step 5.3.3.1.5.6.4.2

Rewrite the expression.

$x = \frac{\sqrt[8]{y} {\sqrt[8]{6}}^{7}}{4 \cdot 6^{1}} + \frac{- 9}{4}$

Step 5.3.3.1.5.6.5

Evaluate the exponent.

$x = \frac{\sqrt[8]{y} {\sqrt[8]{6}}^{7}}{4 \cdot 6} + \frac{- 9}{4}$

Step 5.3.3.1.6

Simplify the numerator.

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Step 5.3.3.1.6.1

Rewrite ${\sqrt[8]{6}}^{7}$ as $\sqrt[8]{6^{7}}$ .

$x = \frac{\sqrt[8]{y} \sqrt[8]{6^{7}}}{4 \cdot 6} + \frac{- 9}{4}$

Step 5.3.3.1.6.2

Raise $6$ to the power of $7$ .

$x = \frac{\sqrt[8]{y} \sqrt[8]{279936}}{4 \cdot 6} + \frac{- 9}{4}$

Step 5.3.3.1.7

Multiply $4$ by $6$ .

$x = \frac{\sqrt[8]{y} \sqrt[8]{279936}}{24} + \frac{- 9}{4}$

Step 5.3.3.1.8

Combine using the product rule for radicals.

$x = \frac{\sqrt[8]{y \cdot 279936}}{24} + \frac{- 9}{4}$

Step 5.3.3.1.9

Move the negative in front of the fraction.

$x = \frac{\sqrt[8]{y \cdot 279936}}{24} - \frac{9}{4}$

Step 5.3.3.2

Reorder factors in $\frac{\sqrt[8]{y \cdot 279936}}{24} - \frac{9}{4}$ .

$x = \frac{\sqrt[8]{279936 y}}{24} - \frac{9}{4}$

Step 5.4

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

$4 x + 9 = - \frac{\sqrt[8]{y}}{\sqrt[8]{6}}$

Step 5.5

Subtract $9$ from both sides of the equation.

$4 x = - \frac{\sqrt[8]{y}}{\sqrt[8]{6}} - 9$

Step 5.6

Divide each term in $4 x = - \frac{\sqrt[8]{y}}{\sqrt[8]{6}} - 9$ by $4$ and simplify.

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Step 5.6.1

Divide each term in $4 x = - \frac{\sqrt[8]{y}}{\sqrt[8]{6}} - 9$ by $4$ .

$\frac{4 x}{4} = \frac{- \frac{\sqrt[8]{y}}{\sqrt[8]{6}}}{4} + \frac{- 9}{4}$

Step 5.6.2

Simplify the left side.

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Step 5.6.2.1

Cancel the common factor of $4$ .

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Step 5.6.2.1.1

Cancel the common factor.

$\frac{4 x}{4} = \frac{- \frac{\sqrt[8]{y}}{\sqrt[8]{6}}}{4} + \frac{- 9}{4}$

Step 5.6.2.1.2

Divide $x$ by $1$ .

$x = \frac{- \frac{\sqrt[8]{y}}{\sqrt[8]{6}}}{4} + \frac{- 9}{4}$

Step 5.6.3

Simplify the right side.

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Step 5.6.3.1

Simplify each term.

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Step 5.6.3.1.1

Multiply the numerator by the reciprocal of the denominator.

$x = - \frac{\sqrt[8]{y}}{\sqrt[8]{6}} \cdot \frac{1}{4} + \frac{- 9}{4}$

Step 5.6.3.1.2

Multiply $\frac{1}{4}$ by $\frac{\sqrt[8]{y}}{\sqrt[8]{6}}$ .

$x = - \frac{\sqrt[8]{y}}{4 \sqrt[8]{6}} + \frac{- 9}{4}$

Step 5.6.3.1.3

Multiply $\frac{\sqrt[8]{y}}{4 \sqrt[8]{6}}$ by $\frac{{\sqrt[8]{6}}^{7}}{{\sqrt[8]{6}}^{7}}$ .

$x = - (\frac{\sqrt[8]{y}}{4 \sqrt[8]{6}} \cdot \frac{{\sqrt[8]{6}}^{7}}{{\sqrt[8]{6}}^{7}}) + \frac{- 9}{4}$

Step 5.6.3.1.4

Combine and simplify the denominator.

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Step 5.6.3.1.4.1

Multiply $\frac{\sqrt[8]{y}}{4 \sqrt[8]{6}}$ by $\frac{{\sqrt[8]{6}}^{7}}{{\sqrt[8]{6}}^{7}}$ .

$x = - \frac{\sqrt[8]{y} {\sqrt[8]{6}}^{7}}{4 \sqrt[8]{6} {\sqrt[8]{6}}^{7}} + \frac{- 9}{4}$

Step 5.6.3.1.4.2

Move $\sqrt[8]{6}$ .

$x = - \frac{\sqrt[8]{y} {\sqrt[8]{6}}^{7}}{4 (\sqrt[8]{6} {\sqrt[8]{6}}^{7})} + \frac{- 9}{4}$

Step 5.6.3.1.4.3

Raise $\sqrt[8]{6}$ to the power of $1$ .

$x = - \frac{\sqrt[8]{y} {\sqrt[8]{6}}^{7}}{4 ({\sqrt[8]{6}}^{1} {\sqrt[8]{6}}^{7})} + \frac{- 9}{4}$

Step 5.6.3.1.4.4

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

$x = - \frac{\sqrt[8]{y} {\sqrt[8]{6}}^{7}}{4 {\sqrt[8]{6}}^{1 + 7}} + \frac{- 9}{4}$

Step 5.6.3.1.4.5

Add $1$ and $7$ .

$x = - \frac{\sqrt[8]{y} {\sqrt[8]{6}}^{7}}{4 {\sqrt[8]{6}}^{8}} + \frac{- 9}{4}$

Step 5.6.3.1.4.6

Rewrite ${\sqrt[8]{6}}^{8}$ as $6$ .

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Step 5.6.3.1.4.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[8]{6}$ as $6^{\frac{1}{8}}$ .

$x = - \frac{\sqrt[8]{y} {\sqrt[8]{6}}^{7}}{4 {(6^{\frac{1}{8}})}^{8}} + \frac{- 9}{4}$

Step 5.6.3.1.4.6.2

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

$x = - \frac{\sqrt[8]{y} {\sqrt[8]{6}}^{7}}{4 \cdot 6^{\frac{1}{8} \cdot 8}} + \frac{- 9}{4}$

Step 5.6.3.1.4.6.3

Combine $\frac{1}{8}$ and $8$ .

$x = - \frac{\sqrt[8]{y} {\sqrt[8]{6}}^{7}}{4 \cdot 6^{\frac{8}{8}}} + \frac{- 9}{4}$

Step 5.6.3.1.4.6.4

Cancel the common factor of $8$ .

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Step 5.6.3.1.4.6.4.1

Cancel the common factor.

$x = - \frac{\sqrt[8]{y} {\sqrt[8]{6}}^{7}}{4 \cdot 6^{\frac{8}{8}}} + \frac{- 9}{4}$

Step 5.6.3.1.4.6.4.2

Rewrite the expression.

$x = - \frac{\sqrt[8]{y} {\sqrt[8]{6}}^{7}}{4 \cdot 6^{1}} + \frac{- 9}{4}$

Step 5.6.3.1.4.6.5

Evaluate the exponent.

$x = - \frac{\sqrt[8]{y} {\sqrt[8]{6}}^{7}}{4 \cdot 6} + \frac{- 9}{4}$

Step 5.6.3.1.5

Simplify the numerator.

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Step 5.6.3.1.5.1

Rewrite ${\sqrt[8]{6}}^{7}$ as $\sqrt[8]{6^{7}}$ .

$x = - \frac{\sqrt[8]{y} \sqrt[8]{6^{7}}}{4 \cdot 6} + \frac{- 9}{4}$

Step 5.6.3.1.5.2

Raise $6$ to the power of $7$ .

$x = - \frac{\sqrt[8]{y} \sqrt[8]{279936}}{4 \cdot 6} + \frac{- 9}{4}$

Step 5.6.3.1.6

Multiply $4$ by $6$ .

$x = - \frac{\sqrt[8]{y} \sqrt[8]{279936}}{24} + \frac{- 9}{4}$

Step 5.6.3.1.7

Combine using the product rule for radicals.

$x = - \frac{\sqrt[8]{y \cdot 279936}}{24} + \frac{- 9}{4}$

Step 5.6.3.1.8

Move the negative in front of the fraction.

$x = - \frac{\sqrt[8]{y \cdot 279936}}{24} - \frac{9}{4}$

Step 5.6.3.2

Reorder factors in $- \frac{\sqrt[8]{y \cdot 279936}}{24} - \frac{9}{4}$ .

$x = - \frac{\sqrt[8]{279936 y}}{24} - \frac{9}{4}$

Step 5.7

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

$x = \frac{\sqrt[8]{279936 y}}{24} - \frac{9}{4}$

$x = - \frac{\sqrt[8]{279936 y}}{24} - \frac{9}{4}$

$x = \frac{\sqrt[8]{279936 y}}{24} - \frac{9}{4}$

$x = - \frac{\sqrt[8]{279936 y}}{24} - \frac{9}{4}$