Precalculus Examples

$\frac{11}{6} = {(9 x + 7)}^{\frac{8}{11}}$

Step 1

Rewrite the equation as ${(9 x + 7)}^{\frac{8}{11}} = \frac{11}{6}$ .

${(9 x + 7)}^{\frac{8}{11}} = \frac{11}{6}$

Step 2

Raise each side of the equation to the power of $\frac{11}{8}$ to eliminate the fractional exponent on the left side.

${({(9 x + 7)}^{\frac{8}{11}})}^{\frac{11}{8}} = \pm {(\frac{11}{6})}^{\frac{11}{8}}$

Step 3

Simplify the exponent.

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Step 3.1

Simplify the left side.

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Step 3.1.1

Simplify ${({(9 x + 7)}^{\frac{8}{11}})}^{\frac{11}{8}}$ .

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Step 3.1.1.1

Multiply the exponents in ${({(9 x + 7)}^{\frac{8}{11}})}^{\frac{11}{8}}$ .

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Step 3.1.1.1.1

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

${(9 x + 7)}^{\frac{8}{11} \cdot \frac{11}{8}} = \pm {(\frac{11}{6})}^{\frac{11}{8}}$

Step 3.1.1.1.2

Cancel the common factor of $8$ .

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Step 3.1.1.1.2.1

Cancel the common factor.

${(9 x + 7)}^{\frac{8}{11} \cdot \frac{11}{8}} = \pm {(\frac{11}{6})}^{\frac{11}{8}}$

Step 3.1.1.1.2.2

Rewrite the expression.

${(9 x + 7)}^{\frac{1}{11} \cdot 11} = \pm {(\frac{11}{6})}^{\frac{11}{8}}$

Step 3.1.1.1.3

Cancel the common factor of $11$ .

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Step 3.1.1.1.3.1

Cancel the common factor.

${(9 x + 7)}^{\frac{1}{11} \cdot 11} = \pm {(\frac{11}{6})}^{\frac{11}{8}}$

Step 3.1.1.1.3.2

Rewrite the expression.

${(9 x + 7)}^{1} = \pm {(\frac{11}{6})}^{\frac{11}{8}}$

Step 3.1.1.2

Simplify.

$9 x + 7 = \pm {(\frac{11}{6})}^{\frac{11}{8}}$

Step 3.2

Simplify the right side.

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Step 3.2.1

Apply the product rule to $\frac{11}{6}$ .

$9 x + 7 = \pm \frac{11^{\frac{11}{8}}}{6^{\frac{11}{8}}}$

Step 4

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

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Step 4.1

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

$9 x + 7 = \frac{11^{\frac{11}{8}}}{6^{\frac{11}{8}}}$

Step 4.2

Subtract $7$ from both sides of the equation.

$9 x = \frac{11^{\frac{11}{8}}}{6^{\frac{11}{8}}} - 7$

Step 4.3

Divide each term in $9 x = \frac{11^{\frac{11}{8}}}{6^{\frac{11}{8}}} - 7$ by $9$ and simplify.

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Step 4.3.1

Divide each term in $9 x = \frac{11^{\frac{11}{8}}}{6^{\frac{11}{8}}} - 7$ by $9$ .

$\frac{9 x}{9} = \frac{\frac{11^{\frac{11}{8}}}{6^{\frac{11}{8}}}}{9} + \frac{- 7}{9}$

Step 4.3.2

Simplify the left side.

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Step 4.3.2.1

Cancel the common factor of $9$ .

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Step 4.3.2.1.1

Cancel the common factor.

$\frac{9 x}{9} = \frac{\frac{11^{\frac{11}{8}}}{6^{\frac{11}{8}}}}{9} + \frac{- 7}{9}$

Step 4.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{\frac{11^{\frac{11}{8}}}{6^{\frac{11}{8}}}}{9} + \frac{- 7}{9}$

Step 4.3.3

Simplify the right side.

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Step 4.3.3.1

Simplify each term.

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Step 4.3.3.1.1

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{11^{\frac{11}{8}}}{6^{\frac{11}{8}}} \cdot \frac{1}{9} + \frac{- 7}{9}$

Step 4.3.3.1.2

Combine.

$x = \frac{11^{\frac{11}{8}} \cdot 1}{6^{\frac{11}{8}} \cdot 9} + \frac{- 7}{9}$

Step 4.3.3.1.3

Multiply $11^{\frac{11}{8}}$ by $1$ .

$x = \frac{11^{\frac{11}{8}}}{6^{\frac{11}{8}} \cdot 9} + \frac{- 7}{9}$

Step 4.3.3.1.4

Move $9$ to the left of $6^{\frac{11}{8}}$ .

$x = \frac{11^{\frac{11}{8}}}{9 \cdot 6^{\frac{11}{8}}} + \frac{- 7}{9}$

Step 4.3.3.1.5

Move the negative in front of the fraction.

$x = \frac{11^{\frac{11}{8}}}{9 \cdot 6^{\frac{11}{8}}} - \frac{7}{9}$

Step 4.4

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

$9 x + 7 = - \frac{11^{\frac{11}{8}}}{6^{\frac{11}{8}}}$

Step 4.5

Subtract $7$ from both sides of the equation.

$9 x = - \frac{11^{\frac{11}{8}}}{6^{\frac{11}{8}}} - 7$

Step 4.6

Divide each term in $9 x = - \frac{11^{\frac{11}{8}}}{6^{\frac{11}{8}}} - 7$ by $9$ and simplify.

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Step 4.6.1

Divide each term in $9 x = - \frac{11^{\frac{11}{8}}}{6^{\frac{11}{8}}} - 7$ by $9$ .

$\frac{9 x}{9} = \frac{- \frac{11^{\frac{11}{8}}}{6^{\frac{11}{8}}}}{9} + \frac{- 7}{9}$

Step 4.6.2

Simplify the left side.

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Step 4.6.2.1

Cancel the common factor of $9$ .

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Step 4.6.2.1.1

Cancel the common factor.

$\frac{9 x}{9} = \frac{- \frac{11^{\frac{11}{8}}}{6^{\frac{11}{8}}}}{9} + \frac{- 7}{9}$

Step 4.6.2.1.2

Divide $x$ by $1$ .

$x = \frac{- \frac{11^{\frac{11}{8}}}{6^{\frac{11}{8}}}}{9} + \frac{- 7}{9}$

Step 4.6.3

Simplify the right side.

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Step 4.6.3.1

Simplify each term.

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Step 4.6.3.1.1

Multiply the numerator by the reciprocal of the denominator.

$x = - \frac{11^{\frac{11}{8}}}{6^{\frac{11}{8}}} \cdot \frac{1}{9} + \frac{- 7}{9}$

Step 4.6.3.1.2

Multiply $\frac{1}{9}$ by $\frac{11^{\frac{11}{8}}}{6^{\frac{11}{8}}}$ .

$x = - \frac{11^{\frac{11}{8}}}{9 \cdot 6^{\frac{11}{8}}} + \frac{- 7}{9}$

Step 4.6.3.1.3

Move the negative in front of the fraction.

$x = - \frac{11^{\frac{11}{8}}}{9 \cdot 6^{\frac{11}{8}}} - \frac{7}{9}$

Step 4.7

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

$x = \frac{11^{\frac{11}{8}}}{9 \cdot 6^{\frac{11}{8}}} - \frac{7}{9}, - \frac{11^{\frac{11}{8}}}{9 \cdot 6^{\frac{11}{8}}} - \frac{7}{9}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = \frac{11^{\frac{11}{8}}}{9 \cdot 6^{\frac{11}{8}}} - \frac{7}{9}, - \frac{11^{\frac{11}{8}}}{9 \cdot 6^{\frac{11}{8}}} - \frac{7}{9}$

Decimal Form:

$x = - 0.52208735 \dots, - 1.03346820 \dots$