Precalculus Examples

$7 - {(4 - 2 x)}^{\frac{2}{3}} = 3$

Step 1

Move all terms to the left side of the equation and simplify.

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Step 1.1

Subtract $3$ from both sides of the equation.

$7 - {(4 - 2 x)}^{\frac{2}{3}} - 3 = 0$

Step 1.2

Subtract $3$ from $7$ .

$4 - {(4 - 2 x)}^{\frac{2}{3}} = 0$

Step 2

Subtract $4$ from both sides of the equation.

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

Step 3

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

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

Step 4

Simplify the left side.

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Step 4.1

Simplify ${(- {(4 - 2 x)}^{\frac{2}{3}})}^{\frac{3}{2}}$ .

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Step 4.1.1

Apply the product rule to $- {(4 - 2 x)}^{\frac{2}{3}}$ .

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

Step 4.1.2

Rewrite ${(- 1)}^{\frac{3}{2}}$ as $\sqrt{{(- 1)}^{3}}$ .

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

Step 4.1.3

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

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

Step 4.1.4

Rewrite $\sqrt{- 1}$ as $i$ .

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

Step 4.1.5

Multiply the exponents in ${({(4 - 2 x)}^{\frac{2}{3}})}^{\frac{3}{2}}$ .

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Step 4.1.5.1

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

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

Step 4.1.5.2

Cancel the common factor of $2$ .

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Step 4.1.5.2.1

Cancel the common factor.

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

Step 4.1.5.2.2

Rewrite the expression.

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

Step 4.1.5.3

Cancel the common factor of $3$ .

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Step 4.1.5.3.1

Cancel the common factor.

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

Step 4.1.5.3.2

Rewrite the expression.

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

Step 4.1.6

Simplify.

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

Step 4.1.7

Apply the distributive property.

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

Step 4.1.8

Simplify the expression.

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Step 4.1.8.1

Move $4$ to the left of $i$ .

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

Step 4.1.8.2

Reorder factors in $4 i + i (- 2 x)$ .

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

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.

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

Step 5.2

Subtract $4 i$ from both sides of the equation.

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

Step 5.3

Divide each term in $- 2 i x = {(- 4)}^{\frac{3}{2}} - 4 i$ by $- 2 i$ and simplify.

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Step 5.3.1

Divide each term in $- 2 i x = {(- 4)}^{\frac{3}{2}} - 4 i$ by $- 2 i$ .

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

Step 5.3.2

Simplify the left side.

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Step 5.3.2.1

Cancel the common factor of $- 2$ .

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Step 5.3.2.1.1

Cancel the common factor.

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

Step 5.3.2.1.2

Rewrite the expression.

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

Step 5.3.2.2

Cancel the common factor of $i$ .

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Step 5.3.2.2.1

Cancel the common factor.

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

Step 5.3.2.2.2

Divide $x$ by $1$ .

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

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 and denominator of $\frac{{(- 4)}^{\frac{3}{2}}}{- 2 i}$ by the conjugate of $- 2 i$ to make the denominator real.

$x = \frac{{(- 4)}^{\frac{3}{2}}}{- 2 i} \cdot \frac{i}{i} + \frac{- 4 i}{- 2 i}$

Step 5.3.3.1.2

Multiply.

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Step 5.3.3.1.2.1

Combine.

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

Step 5.3.3.1.2.2

Reorder factors in ${(- 4)}^{\frac{3}{2}} i$ .

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

Step 5.3.3.1.2.3

Simplify the denominator.

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Step 5.3.3.1.2.3.1

Add parentheses.

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

Step 5.3.3.1.2.3.2

Raise $i$ to the power of $1$ .

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

Step 5.3.3.1.2.3.3

Raise $i$ to the power of $1$ .

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

Step 5.3.3.1.2.3.4

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

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

Step 5.3.3.1.2.3.5

Add $1$ and $1$ .

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

Step 5.3.3.1.2.3.6

Rewrite $i^{2}$ as $- 1$ .

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

Step 5.3.3.1.3

Multiply $- 2$ by $- 1$ .

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

Step 5.3.3.1.4

Cancel the common factor of $- 4$ and $- 2$ .

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Step 5.3.3.1.4.1

Factor $- 2$ out of $- 4 i$ .

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

Step 5.3.3.1.4.2

Cancel the common factors.

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Step 5.3.3.1.4.2.1

Factor $- 2$ out of $- 2 i$ .

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

Step 5.3.3.1.4.2.2

Cancel the common factor.

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

Step 5.3.3.1.4.2.3

Rewrite the expression.

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

Step 5.3.3.1.5

Cancel the common factor of $i$ .

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Step 5.3.3.1.5.1

Cancel the common factor.

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

Step 5.3.3.1.5.2

Divide $2$ by $1$ .

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

Step 5.3.3.2

Reorder $\frac{i {(- 4)}^{\frac{3}{2}}}{2}$ and $2$ .

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

Step 5.4

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

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

Step 5.5

Subtract $4 i$ from both sides of the equation.

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

Step 5.6

Divide each term in $- 2 i x = - {(- 4)}^{\frac{3}{2}} - 4 i$ by $- 2 i$ and simplify.

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Step 5.6.1

Divide each term in $- 2 i x = - {(- 4)}^{\frac{3}{2}} - 4 i$ by $- 2 i$ .

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

Step 5.6.2

Simplify the left side.

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Step 5.6.2.1

Cancel the common factor of $- 2$ .

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Step 5.6.2.1.1

Cancel the common factor.

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

Step 5.6.2.1.2

Rewrite the expression.

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

Step 5.6.2.2

Cancel the common factor of $i$ .

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Step 5.6.2.2.1

Cancel the common factor.

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

Step 5.6.2.2.2

Divide $x$ by $1$ .

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

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 and denominator of $\frac{- {(- 4)}^{\frac{3}{2}}}{- 2 i}$ by the conjugate of $- 2 i$ to make the denominator real.

$x = \frac{- {(- 4)}^{\frac{3}{2}}}{- 2 i} \cdot \frac{i}{i} + \frac{- 4 i}{- 2 i}$

Step 5.6.3.1.2

Multiply.

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Step 5.6.3.1.2.1

Combine.

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

Step 5.6.3.1.2.2

Reorder factors in $- {(- 4)}^{\frac{3}{2}} i$ .

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

Step 5.6.3.1.2.3

Simplify the denominator.

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Step 5.6.3.1.2.3.1

Add parentheses.

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

Step 5.6.3.1.2.3.2

Raise $i$ to the power of $1$ .

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

Step 5.6.3.1.2.3.3

Raise $i$ to the power of $1$ .

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

Step 5.6.3.1.2.3.4

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

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

Step 5.6.3.1.2.3.5

Add $1$ and $1$ .

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

Step 5.6.3.1.2.3.6

Rewrite $i^{2}$ as $- 1$ .

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

Step 5.6.3.1.3

Dividing two negative values results in a positive value.

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

Step 5.6.3.1.4

Move the negative in front of the fraction.

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

Step 5.6.3.1.5

Cancel the common factor of $- 4$ and $- 2$ .

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Step 5.6.3.1.5.1

Factor $- 2$ out of $- 4 i$ .

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

Step 5.6.3.1.5.2

Cancel the common factors.

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Step 5.6.3.1.5.2.1

Factor $- 2$ out of $- 2 i$ .

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

Step 5.6.3.1.5.2.2

Cancel the common factor.

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

Step 5.6.3.1.5.2.3

Rewrite the expression.

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

Step 5.6.3.1.6

Cancel the common factor of $i$ .

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Step 5.6.3.1.6.1

Cancel the common factor.

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

Step 5.6.3.1.6.2

Divide $2$ by $1$ .

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

Step 5.6.3.2

Reorder $- \frac{i {(- 4)}^{\frac{3}{2}}}{2}$ and $2$ .

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

Step 5.7

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

$x = 2 + \frac{i {(- 4)}^{\frac{3}{2}}}{2}, 2 - \frac{i {(- 4)}^{\frac{3}{2}}}{2}$