Algebra Examples

$4 {(x - 2)}^{\frac{4}{3}} - 5 = 59$

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.

$4 {(x - 2)}^{\frac{4}{3}} = 59 + 5$

Step 1.2

Add $59$ and $5$ .

$4 {(x - 2)}^{\frac{4}{3}} = 64$

Step 2

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

${(4 {(x - 2)}^{\frac{4}{3}})}^{\frac{3}{4}} = \pm 64^{\frac{3}{4}}$

Step 3

Simplify the left side.

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Step 3.1

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

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Step 3.1.1

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

$4^{\frac{3}{4}} {({(x - 2)}^{\frac{4}{3}})}^{\frac{3}{4}} = \pm 64^{\frac{3}{4}}$

Step 3.1.2

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

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Step 3.1.2.1

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

$4^{\frac{3}{4}} {(x - 2)}^{\frac{4}{3} \cdot \frac{3}{4}} = \pm 64^{\frac{3}{4}}$

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.

$4^{\frac{3}{4}} {(x - 2)}^{\frac{4}{3} \cdot \frac{3}{4}} = \pm 64^{\frac{3}{4}}$

Step 3.1.2.2.2

Rewrite the expression.

$4^{\frac{3}{4}} {(x - 2)}^{\frac{1}{3} \cdot 3} = \pm 64^{\frac{3}{4}}$

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.

$4^{\frac{3}{4}} {(x - 2)}^{\frac{1}{3} \cdot 3} = \pm 64^{\frac{3}{4}}$

Step 3.1.2.3.2

Rewrite the expression.

$4^{\frac{3}{4}} {(x - 2)}^{1} = \pm 64^{\frac{3}{4}}$

Step 3.1.3

Simplify.

$4^{\frac{3}{4}} (x - 2) = \pm 64^{\frac{3}{4}}$

Step 3.1.4

Apply the distributive property.

$4^{\frac{3}{4}} x + 4^{\frac{3}{4}} \cdot - 2 = \pm 64^{\frac{3}{4}}$

Step 3.1.5

Move $- 2$ to the left of $4^{\frac{3}{4}}$ .

$4^{\frac{3}{4}} x - 2 \cdot 4^{\frac{3}{4}} = \pm 64^{\frac{3}{4}}$

Step 3.1.6

Multiply $- 2 \cdot 4^{\frac{3}{4}}$ .

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Step 3.1.6.1

Factor out negative.

$4^{\frac{3}{4}} x - (2 \cdot 4^{\frac{3}{4}}) = \pm 64^{\frac{3}{4}}$

Step 3.1.6.2

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

$4^{\frac{3}{4}} x - (2 \cdot {(2^{2})}^{\frac{3}{4}}) = \pm 64^{\frac{3}{4}}$

Step 3.1.6.3

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

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Step 3.1.6.3.1

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

$4^{\frac{3}{4}} x - (2 \cdot 2^{2 (\frac{3}{4})}) = \pm 64^{\frac{3}{4}}$

Step 3.1.6.3.2

Cancel the common factor of $2$ .

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Step 3.1.6.3.2.1

Factor $2$ out of $4$ .

$4^{\frac{3}{4}} x - (2 \cdot 2^{2 \frac{3}{2 (2)}}) = \pm 64^{\frac{3}{4}}$

Step 3.1.6.3.2.2

Cancel the common factor.

$4^{\frac{3}{4}} x - (2 \cdot 2^{2 \frac{3}{2 \cdot 2}}) = \pm 64^{\frac{3}{4}}$

Step 3.1.6.3.2.3

Rewrite the expression.

$4^{\frac{3}{4}} x - (2 \cdot 2^{\frac{3}{2}}) = \pm 64^{\frac{3}{4}}$

Step 3.1.6.4

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

$4^{\frac{3}{4}} x - 2^{1 + \frac{3}{2}} = \pm 64^{\frac{3}{4}}$

Step 3.1.6.5

Write $1$ as a fraction with a common denominator.

$4^{\frac{3}{4}} x - 2^{\frac{2}{2} + \frac{3}{2}} = \pm 64^{\frac{3}{4}}$

Step 3.1.6.6

Combine the numerators over the common denominator.

$4^{\frac{3}{4}} x - 2^{\frac{2 + 3}{2}} = \pm 64^{\frac{3}{4}}$

Step 3.1.6.7

Add $2$ and $3$ .

$4^{\frac{3}{4}} x - 2^{\frac{5}{2}} = \pm 64^{\frac{3}{4}}$

Step 3.1.7

Reorder factors in $4^{\frac{3}{4}} x - 2^{\frac{5}{2}}$ .

$x \cdot 4^{\frac{3}{4}} - 2^{\frac{5}{2}} = \pm 64^{\frac{3}{4}}$

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.

$x \cdot 4^{\frac{3}{4}} - 2^{\frac{5}{2}} = 64^{\frac{3}{4}}$

Step 4.2

Add $2^{\frac{5}{2}}$ to both sides of the equation.

$x \cdot 4^{\frac{3}{4}} = 64^{\frac{3}{4}} + 2^{\frac{5}{2}}$

Step 4.3

Divide each term in $x \cdot 4^{\frac{3}{4}} = 64^{\frac{3}{4}} + 2^{\frac{5}{2}}$ by $4^{\frac{3}{4}}$ and simplify.

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Step 4.3.1

Divide each term in $x \cdot 4^{\frac{3}{4}} = 64^{\frac{3}{4}} + 2^{\frac{5}{2}}$ by $4^{\frac{3}{4}}$ .

$\frac{x \cdot 4^{\frac{3}{4}}}{4^{\frac{3}{4}}} = \frac{64^{\frac{3}{4}}}{4^{\frac{3}{4}}} + \frac{2^{\frac{5}{2}}}{4^{\frac{3}{4}}}$

Step 4.3.2

Simplify the left side.

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Step 4.3.2.1

Cancel the common factor.

$\frac{x \cdot 4^{\frac{3}{4}}}{4^{\frac{3}{4}}} = \frac{64^{\frac{3}{4}}}{4^{\frac{3}{4}}} + \frac{2^{\frac{5}{2}}}{4^{\frac{3}{4}}}$

Step 4.3.2.2

Divide $x$ by $1$ .

$x = \frac{64^{\frac{3}{4}}}{4^{\frac{3}{4}}} + \frac{2^{\frac{5}{2}}}{4^{\frac{3}{4}}}$

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

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

$x = {(\frac{64}{4})}^{\frac{3}{4}} + \frac{2^{\frac{5}{2}}}{4^{\frac{3}{4}}}$

Step 4.3.3.1.2

Divide $64$ by $4$ .

$x = 16^{\frac{3}{4}} + \frac{2^{\frac{5}{2}}}{4^{\frac{3}{4}}}$

Step 4.3.3.1.3

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

$x = {(2^{4})}^{\frac{3}{4}} + \frac{2^{\frac{5}{2}}}{4^{\frac{3}{4}}}$

Step 4.3.3.1.4

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

$x = 2^{4 (\frac{3}{4})} + \frac{2^{\frac{5}{2}}}{4^{\frac{3}{4}}}$

Step 4.3.3.1.5

Cancel the common factor of $4$ .

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Step 4.3.3.1.5.1

Cancel the common factor.

$x = 2^{4 (\frac{3}{4})} + \frac{2^{\frac{5}{2}}}{4^{\frac{3}{4}}}$

Step 4.3.3.1.5.2

Rewrite the expression.

$x = 2^{3} + \frac{2^{\frac{5}{2}}}{4^{\frac{3}{4}}}$

Step 4.3.3.1.6

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

$x = 8 + \frac{2^{\frac{5}{2}}}{4^{\frac{3}{4}}}$

Step 4.4

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

$x \cdot 4^{\frac{3}{4}} - 2^{\frac{5}{2}} = - 64^{\frac{3}{4}}$

Step 4.5

Add $2^{\frac{5}{2}}$ to both sides of the equation.

$x \cdot 4^{\frac{3}{4}} = - 64^{\frac{3}{4}} + 2^{\frac{5}{2}}$

Step 4.6

Divide each term in $x \cdot 4^{\frac{3}{4}} = - 64^{\frac{3}{4}} + 2^{\frac{5}{2}}$ by $4^{\frac{3}{4}}$ and simplify.

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Step 4.6.1

Divide each term in $x \cdot 4^{\frac{3}{4}} = - 64^{\frac{3}{4}} + 2^{\frac{5}{2}}$ by $4^{\frac{3}{4}}$ .

$\frac{x \cdot 4^{\frac{3}{4}}}{4^{\frac{3}{4}}} = \frac{- 64^{\frac{3}{4}}}{4^{\frac{3}{4}}} + \frac{2^{\frac{5}{2}}}{4^{\frac{3}{4}}}$

Step 4.6.2

Simplify the left side.

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Step 4.6.2.1

Cancel the common factor.

$\frac{x \cdot 4^{\frac{3}{4}}}{4^{\frac{3}{4}}} = \frac{- 64^{\frac{3}{4}}}{4^{\frac{3}{4}}} + \frac{2^{\frac{5}{2}}}{4^{\frac{3}{4}}}$

Step 4.6.2.2

Divide $x$ by $1$ .

$x = \frac{- 64^{\frac{3}{4}}}{4^{\frac{3}{4}}} + \frac{2^{\frac{5}{2}}}{4^{\frac{3}{4}}}$

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

Move the negative in front of the fraction.

$x = - \frac{64^{\frac{3}{4}}}{4^{\frac{3}{4}}} + \frac{2^{\frac{5}{2}}}{4^{\frac{3}{4}}}$

Step 4.6.3.1.2

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

$x = - {(\frac{64}{4})}^{\frac{3}{4}} + \frac{2^{\frac{5}{2}}}{4^{\frac{3}{4}}}$

Step 4.6.3.1.3

Divide $64$ by $4$ .

$x = - 16^{\frac{3}{4}} + \frac{2^{\frac{5}{2}}}{4^{\frac{3}{4}}}$

Step 4.6.3.1.4

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

$x = - {(2^{4})}^{\frac{3}{4}} + \frac{2^{\frac{5}{2}}}{4^{\frac{3}{4}}}$

Step 4.6.3.1.5

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

$x = - 2^{4 (\frac{3}{4})} + \frac{2^{\frac{5}{2}}}{4^{\frac{3}{4}}}$

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.

$x = - 2^{4 (\frac{3}{4})} + \frac{2^{\frac{5}{2}}}{4^{\frac{3}{4}}}$

Step 4.6.3.1.6.2

Rewrite the expression.

$x = - 2^{3} + \frac{2^{\frac{5}{2}}}{4^{\frac{3}{4}}}$

Step 4.6.3.1.7

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

$x = - 1 \cdot 8 + \frac{2^{\frac{5}{2}}}{4^{\frac{3}{4}}}$

Step 4.6.3.1.8

Multiply $- 1$ by $8$ .

$x = - 8 + \frac{2^{\frac{5}{2}}}{4^{\frac{3}{4}}}$

Step 4.7

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

$x = 8 + \frac{2^{\frac{5}{2}}}{4^{\frac{3}{4}}}, - 8 + \frac{2^{\frac{5}{2}}}{4^{\frac{3}{4}}}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = 8 + \frac{2^{\frac{5}{2}}}{4^{\frac{3}{4}}}, - 8 + \frac{2^{\frac{5}{2}}}{4^{\frac{3}{4}}}$

Decimal Form:

$x = 10, - 6$