Precalculus Examples

${(2 x - 1)}^{\frac{2}{3}} = 4$

Step 1

Raise each side of the equation to the power of $\frac{3}{2}$ to eliminate the fractional exponent on the left side.

${({(2 x - 1)}^{\frac{2}{3}})}^{\frac{3}{2}} = \pm 4^{\frac{3}{2}}$

Step 2

Simplify the exponent.

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Step 2.1

Simplify the left side.

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Step 2.1.1

Simplify ${({(2 x - 1)}^{\frac{2}{3}})}^{\frac{3}{2}}$ .

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Step 2.1.1.1

Multiply the exponents in ${({(2 x - 1)}^{\frac{2}{3}})}^{\frac{3}{2}}$ .

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Step 2.1.1.1.1

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

${(2 x - 1)}^{\frac{2}{3} \cdot \frac{3}{2}} = \pm 4^{\frac{3}{2}}$

Step 2.1.1.1.2

Cancel the common factor of $2$ .

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Step 2.1.1.1.2.1

Cancel the common factor.

${(2 x - 1)}^{\frac{2}{3} \cdot \frac{3}{2}} = \pm 4^{\frac{3}{2}}$

Step 2.1.1.1.2.2

Rewrite the expression.

${(2 x - 1)}^{\frac{1}{3} \cdot 3} = \pm 4^{\frac{3}{2}}$

Step 2.1.1.1.3

Cancel the common factor of $3$ .

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Step 2.1.1.1.3.1

Cancel the common factor.

${(2 x - 1)}^{\frac{1}{3} \cdot 3} = \pm 4^{\frac{3}{2}}$

Step 2.1.1.1.3.2

Rewrite the expression.

${(2 x - 1)}^{1} = \pm 4^{\frac{3}{2}}$

Step 2.1.1.2

Simplify.

$2 x - 1 = \pm 4^{\frac{3}{2}}$

Step 2.2

Simplify the right side.

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Step 2.2.1

Simplify $\pm 4^{\frac{3}{2}}$ .

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Step 2.2.1.1

Simplify the expression.

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Step 2.2.1.1.1

Rewrite $4$ as $2^{2}$ .

$2 x - 1 = \pm {(2^{2})}^{\frac{3}{2}}$

Step 2.2.1.1.2

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

$2 x - 1 = \pm 2^{2 (\frac{3}{2})}$

Step 2.2.1.2

Cancel the common factor of $2$ .

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Step 2.2.1.2.1

Cancel the common factor.

$2 x - 1 = \pm 2^{2 (\frac{3}{2})}$

Step 2.2.1.2.2

Rewrite the expression.

$2 x - 1 = \pm 2^{3}$

Step 2.2.1.3

Raise $2$ to the power of $3$ .

$2 x - 1 = \pm 8$

Step 3

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

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Step 3.1

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

$2 x - 1 = 8$

Step 3.2

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

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Step 3.2.1

Add $1$ to both sides of the equation.

$2 x = 8 + 1$

Step 3.2.2

Add $8$ and $1$ .

$2 x = 9$

Step 3.3

Divide each term in $2 x = 9$ by $2$ and simplify.

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Step 3.3.1

Divide each term in $2 x = 9$ by $2$ .

$\frac{2 x}{2} = \frac{9}{2}$

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Cancel the common factor of $2$ .

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Step 3.3.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{9}{2}$

Step 3.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{9}{2}$

Step 3.4

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

$2 x - 1 = - 8$

Step 3.5

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

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Step 3.5.1

Add $1$ to both sides of the equation.

$2 x = - 8 + 1$

Step 3.5.2

Add $- 8$ and $1$ .

$2 x = - 7$

Step 3.6

Divide each term in $2 x = - 7$ by $2$ and simplify.

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Step 3.6.1

Divide each term in $2 x = - 7$ by $2$ .

$\frac{2 x}{2} = \frac{- 7}{2}$

Step 3.6.2

Simplify the left side.

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Step 3.6.2.1

Cancel the common factor of $2$ .

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Step 3.6.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{- 7}{2}$

Step 3.6.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 7}{2}$

Step 3.6.3

Simplify the right side.

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Step 3.6.3.1

Move the negative in front of the fraction.

$x = - \frac{7}{2}$

Step 3.7

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

$x = \frac{9}{2}, - \frac{7}{2}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$x = \frac{9}{2}, - \frac{7}{2}$

Decimal Form:

$x = 4.5, - 3.5$

Mixed Number Form:

$x = 4 \frac{1}{2}, - 3 \frac{1}{2}$