Algebra Examples

$4 {(3 - x)}^{\frac{4}{3}} - 5 = 59$

Step 1

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

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Step 1.1

Add $5$ to both sides of the equation.

$4 {(3 - x)}^{\frac{4}{3}} = 59 + 5$

Step 1.2

Add $59$ and $5$ .

$4 {(3 - x)}^{\frac{4}{3}} = 64$

Step 2

Raise each side of the equation to the power of $\frac{3}{4}$ to eliminate the fractional exponent on the left side.

${(4 {(3 - x)}^{\frac{4}{3}})}^{\frac{3}{4}} = \pm 64^{\frac{3}{4}}$

Step 3

Simplify the left side.

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Step 3.1

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

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Step 3.1.1

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

$4^{\frac{3}{4}} {({(3 - x)}^{\frac{4}{3}})}^{\frac{3}{4}} = \pm 64^{\frac{3}{4}}$

Step 3.1.2

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

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Step 3.1.2.1

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

$4^{\frac{3}{4}} {(3 - x)}^{\frac{4}{3} \cdot \frac{3}{4}} = \pm 64^{\frac{3}{4}}$

Step 3.1.2.2

Cancel the common factor of $4$ .

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Step 3.1.2.2.1

Cancel the common factor.

$4^{\frac{3}{4}} {(3 - x)}^{\frac{4}{3} \cdot \frac{3}{4}} = \pm 64^{\frac{3}{4}}$

Step 3.1.2.2.2

Rewrite the expression.

$4^{\frac{3}{4}} {(3 - x)}^{\frac{1}{3} \cdot 3} = \pm 64^{\frac{3}{4}}$

Step 3.1.2.3

Cancel the common factor of $3$ .

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Step 3.1.2.3.1

Cancel the common factor.

$4^{\frac{3}{4}} {(3 - x)}^{\frac{1}{3} \cdot 3} = \pm 64^{\frac{3}{4}}$

Step 3.1.2.3.2

Rewrite the expression.

$4^{\frac{3}{4}} {(3 - x)}^{1} = \pm 64^{\frac{3}{4}}$

Step 3.1.3

Simplify.

$4^{\frac{3}{4}} (3 - x) = \pm 64^{\frac{3}{4}}$

Step 3.1.4

Apply the distributive property.

$4^{\frac{3}{4}} \cdot 3 + 4^{\frac{3}{4}} (- x) = \pm 64^{\frac{3}{4}}$

Step 3.1.5

Reorder.

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Step 3.1.5.1

Move $3$ to the left of $4^{\frac{3}{4}}$ .

$3 \cdot 4^{\frac{3}{4}} + 4^{\frac{3}{4}} (- x) = \pm 64^{\frac{3}{4}}$

Step 3.1.5.2

Reorder factors in $3 \cdot 4^{\frac{3}{4}} + 4^{\frac{3}{4}} (- x)$ .

$3 \cdot 4^{\frac{3}{4}} - x \cdot 4^{\frac{3}{4}} = \pm 64^{\frac{3}{4}}$

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.

$3 \cdot 4^{\frac{3}{4}} - x \cdot 4^{\frac{3}{4}} = 64^{\frac{3}{4}}$

Step 4.2

Subtract $3 \cdot 4^{\frac{3}{4}}$ from both sides of the equation.

$- x \cdot 4^{\frac{3}{4}} = 64^{\frac{3}{4}} - 3 \cdot 4^{\frac{3}{4}}$

Step 4.3

Divide each term in $- x \cdot 4^{\frac{3}{4}} = 64^{\frac{3}{4}} - 3 \cdot 4^{\frac{3}{4}}$ by $- 4^{\frac{3}{4}}$ and simplify.

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Step 4.3.1

Divide each term in $- x \cdot 4^{\frac{3}{4}} = 64^{\frac{3}{4}} - 3 \cdot 4^{\frac{3}{4}}$ by $- 4^{\frac{3}{4}}$ .

$\frac{- x \cdot 4^{\frac{3}{4}}}{- 4^{\frac{3}{4}}} = \frac{64^{\frac{3}{4}}}{- 4^{\frac{3}{4}}} + \frac{- 3 \cdot 4^{\frac{3}{4}}}{- 4^{\frac{3}{4}}}$

Step 4.3.2

Simplify the left side.

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Step 4.3.2.1

Dividing two negative values results in a positive value.

$\frac{x \cdot 4^{\frac{3}{4}}}{4^{\frac{3}{4}}} = \frac{64^{\frac{3}{4}}}{- 4^{\frac{3}{4}}} + \frac{- 3 \cdot 4^{\frac{3}{4}}}{- 4^{\frac{3}{4}}}$

Step 4.3.2.2

Cancel the common factor.

$\frac{x \cdot 4^{\frac{3}{4}}}{4^{\frac{3}{4}}} = \frac{64^{\frac{3}{4}}}{- 4^{\frac{3}{4}}} + \frac{- 3 \cdot 4^{\frac{3}{4}}}{- 4^{\frac{3}{4}}}$

Step 4.3.2.3

Divide $x$ by $1$ .

$x = \frac{64^{\frac{3}{4}}}{- 4^{\frac{3}{4}}} + \frac{- 3 \cdot 4^{\frac{3}{4}}}{- 4^{\frac{3}{4}}}$

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

Move the negative in front of the fraction.

$x = - \frac{64^{\frac{3}{4}}}{4^{\frac{3}{4}}} + \frac{- 3 \cdot 4^{\frac{3}{4}}}{- 4^{\frac{3}{4}}}$

Step 4.3.3.1.2

Use the power of quotient rule $\frac{a^{m}}{b^{m}} = {(\frac{a}{b})}^{m}$ .

$x = - {(\frac{64}{4})}^{\frac{3}{4}} + \frac{- 3 \cdot 4^{\frac{3}{4}}}{- 4^{\frac{3}{4}}}$

Step 4.3.3.1.3

Divide $64$ by $4$ .

$x = - 16^{\frac{3}{4}} + \frac{- 3 \cdot 4^{\frac{3}{4}}}{- 4^{\frac{3}{4}}}$

Step 4.3.3.1.4

Rewrite $16$ as $2^{4}$ .

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

Step 4.3.3.1.5

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

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

Step 4.3.3.1.6

Cancel the common factor of $4$ .

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Step 4.3.3.1.6.1

Cancel the common factor.

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

Step 4.3.3.1.6.2

Rewrite the expression.

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

Step 4.3.3.1.7

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

$x = - 1 \cdot 8 + \frac{- 3 \cdot 4^{\frac{3}{4}}}{- 4^{\frac{3}{4}}}$

Step 4.3.3.1.8

Multiply $- 1$ by $8$ .

$x = - 8 + \frac{- 3 \cdot 4^{\frac{3}{4}}}{- 4^{\frac{3}{4}}}$

Step 4.3.3.1.9

Cancel the common factor.

$x = - 8 + \frac{- 3 \cdot 4^{\frac{3}{4}}}{- 4^{\frac{3}{4}}}$

Step 4.3.3.1.10

Rewrite the expression.

$x = - 8 + \frac{- 3}{- 1}$

Step 4.3.3.1.11

Move the negative one from the denominator of $\frac{- 3}{- 1}$ .

$x = - 8 - 1 \cdot - 3$

Step 4.3.3.1.12

Rewrite $- 1 \cdot - 3$ as $- - 3$ .

$x = - 8 - - 3$

Step 4.3.3.1.13

Multiply $- 1$ by $- 3$ .

$x = - 8 + 3$

Step 4.3.3.2

Add $- 8$ and $3$ .

$x = - 5$

Step 4.4

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

$3 \cdot 4^{\frac{3}{4}} - x \cdot 4^{\frac{3}{4}} = - 64^{\frac{3}{4}}$

Step 4.5

Subtract $3 \cdot 4^{\frac{3}{4}}$ from both sides of the equation.

$- x \cdot 4^{\frac{3}{4}} = - 64^{\frac{3}{4}} - 3 \cdot 4^{\frac{3}{4}}$

Step 4.6

Divide each term in $- x \cdot 4^{\frac{3}{4}} = - 64^{\frac{3}{4}} - 3 \cdot 4^{\frac{3}{4}}$ by $- 4^{\frac{3}{4}}$ and simplify.

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Step 4.6.1

Divide each term in $- x \cdot 4^{\frac{3}{4}} = - 64^{\frac{3}{4}} - 3 \cdot 4^{\frac{3}{4}}$ by $- 4^{\frac{3}{4}}$ .

$\frac{- x \cdot 4^{\frac{3}{4}}}{- 4^{\frac{3}{4}}} = \frac{- 64^{\frac{3}{4}}}{- 4^{\frac{3}{4}}} + \frac{- 3 \cdot 4^{\frac{3}{4}}}{- 4^{\frac{3}{4}}}$

Step 4.6.2

Simplify the left side.

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Step 4.6.2.1

Dividing two negative values results in a positive value.

$\frac{x \cdot 4^{\frac{3}{4}}}{4^{\frac{3}{4}}} = \frac{- 64^{\frac{3}{4}}}{- 4^{\frac{3}{4}}} + \frac{- 3 \cdot 4^{\frac{3}{4}}}{- 4^{\frac{3}{4}}}$

Step 4.6.2.2

Cancel the common factor.

$\frac{x \cdot 4^{\frac{3}{4}}}{4^{\frac{3}{4}}} = \frac{- 64^{\frac{3}{4}}}{- 4^{\frac{3}{4}}} + \frac{- 3 \cdot 4^{\frac{3}{4}}}{- 4^{\frac{3}{4}}}$

Step 4.6.2.3

Divide $x$ by $1$ .

$x = \frac{- 64^{\frac{3}{4}}}{- 4^{\frac{3}{4}}} + \frac{- 3 \cdot 4^{\frac{3}{4}}}{- 4^{\frac{3}{4}}}$

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

Dividing two negative values results in a positive value.

$x = \frac{64^{\frac{3}{4}}}{4^{\frac{3}{4}}} + \frac{- 3 \cdot 4^{\frac{3}{4}}}{- 4^{\frac{3}{4}}}$

Step 4.6.3.1.2

Use the power of quotient rule $\frac{a^{m}}{b^{m}} = {(\frac{a}{b})}^{m}$ .

$x = {(\frac{64}{4})}^{\frac{3}{4}} + \frac{- 3 \cdot 4^{\frac{3}{4}}}{- 4^{\frac{3}{4}}}$

Step 4.6.3.1.3

Divide $64$ by $4$ .

$x = 16^{\frac{3}{4}} + \frac{- 3 \cdot 4^{\frac{3}{4}}}{- 4^{\frac{3}{4}}}$

Step 4.6.3.1.4

Rewrite $16$ as $2^{4}$ .

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

Step 4.6.3.1.5

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

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

Step 4.6.3.1.6

Cancel the common factor of $4$ .

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Step 4.6.3.1.6.1

Cancel the common factor.

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

Step 4.6.3.1.6.2

Rewrite the expression.

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

Step 4.6.3.1.7

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

$x = 8 + \frac{- 3 \cdot 4^{\frac{3}{4}}}{- 4^{\frac{3}{4}}}$

Step 4.6.3.1.8

Cancel the common factor.

$x = 8 + \frac{- 3 \cdot 4^{\frac{3}{4}}}{- 4^{\frac{3}{4}}}$

Step 4.6.3.1.9

Rewrite the expression.

$x = 8 + \frac{- 3}{- 1}$

Step 4.6.3.1.10

Move the negative one from the denominator of $\frac{- 3}{- 1}$ .

$x = 8 - 1 \cdot - 3$

Step 4.6.3.1.11

Rewrite $- 1 \cdot - 3$ as $- - 3$ .

$x = 8 - - 3$

Step 4.6.3.1.12

Multiply $- 1$ by $- 3$ .

$x = 8 + 3$

Step 4.6.3.2

Add $8$ and $3$ .

$x = 11$

Step 4.7

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

$x = - 5, 11$