Algebra Examples

${(x^{2} - x - 4)}^{\frac{3}{4}} - 2 = 6$

Step 1

Subtract $6$ from both sides of the equation.

${(x^{2} - x - 4)}^{\frac{3}{4}} - 2 - 6 = 0$

Step 2

Subtract $6$ from $- 2$ .

${(x^{2} - x - 4)}^{\frac{3}{4}} - 8 = 0$

Step 3

Rewrite ${(x^{2} - x - 4)}^{\frac{3}{4}}$ as ${({(x^{2} - x - 4)}^{\frac{1}{4}})}^{3}$ .

${({(x^{2} - x - 4)}^{\frac{1}{4}})}^{3} - 8 = 0$

Step 4

Rewrite $8$ as $2^{3}$ .

${({(x^{2} - x - 4)}^{\frac{1}{4}})}^{3} - 2^{3} = 0$

Step 5

Since both terms are perfect cubes, factor using the difference of cubes formula, $a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2})$ where $a = {(x^{2} - x - 4)}^{\frac{1}{4}}$ and $b = 2$ .

$({(x^{2} - x - 4)}^{\frac{1}{4}} - 2) ({({(x^{2} - x - 4)}^{\frac{1}{4}})}^{2} + {(x^{2} - x - 4)}^{\frac{1}{4}} \cdot 2 + 2^{2}) = 0$

Step 6

Simplify.

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Step 6.1

Multiply the exponents in ${({(x^{2} - x - 4)}^{\frac{1}{4}})}^{2}$ .

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Step 6.1.1

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

$({(x^{2} - x - 4)}^{\frac{1}{4}} - 2) ({(x^{2} - x - 4)}^{\frac{1}{4} \cdot 2} + {(x^{2} - x - 4)}^{\frac{1}{4}} \cdot 2 + 2^{2}) = 0$

Step 6.1.2

Cancel the common factor of $2$ .

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Step 6.1.2.1

Factor $2$ out of $4$ .

$({(x^{2} - x - 4)}^{\frac{1}{4}} - 2) ({(x^{2} - x - 4)}^{\frac{1}{2 (2)} \cdot 2} + {(x^{2} - x - 4)}^{\frac{1}{4}} \cdot 2 + 2^{2}) = 0$

Step 6.1.2.2

Cancel the common factor.

$({(x^{2} - x - 4)}^{\frac{1}{4}} - 2) ({(x^{2} - x - 4)}^{\frac{1}{2 \cdot 2} \cdot 2} + {(x^{2} - x - 4)}^{\frac{1}{4}} \cdot 2 + 2^{2}) = 0$

Step 6.1.2.3

Rewrite the expression.

$({(x^{2} - x - 4)}^{\frac{1}{4}} - 2) ({(x^{2} - x - 4)}^{\frac{1}{2}} + {(x^{2} - x - 4)}^{\frac{1}{4}} \cdot 2 + 2^{2}) = 0$

Step 6.2

Move $2$ to the left of ${(x^{2} - x - 4)}^{\frac{1}{4}}$ .

$({(x^{2} - x - 4)}^{\frac{1}{4}} - 2) ({(x^{2} - x - 4)}^{\frac{1}{2}} + 2 {(x^{2} - x - 4)}^{\frac{1}{4}} + 2^{2}) = 0$

Step 6.3

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

$({(x^{2} - x - 4)}^{\frac{1}{4}} - 2) ({(x^{2} - x - 4)}^{\frac{1}{2}} + 2 {(x^{2} - x - 4)}^{\frac{1}{4}} + 4) = 0$

Step 6.4

Reorder terms.

$({(x^{2} - x - 4)}^{\frac{1}{4}} - 2) (2 {(x^{2} - x - 4)}^{\frac{1}{4}} + {(x^{2} - x - 4)}^{\frac{1}{2}} + 4) = 0$

Step 7

Simplify $({(x^{2} - x - 4)}^{\frac{1}{4}} - 2) (2 {(x^{2} - x - 4)}^{\frac{1}{4}} + {(x^{2} - x - 4)}^{\frac{1}{2}} + 4)$ .

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Step 7.1

Expand $({(x^{2} - x - 4)}^{\frac{1}{4}} - 2) (2 {(x^{2} - x - 4)}^{\frac{1}{4}} + {(x^{2} - x - 4)}^{\frac{1}{2}} + 4)$ by multiplying each term in the first expression by each term in the second expression.

${(x^{2} - x - 4)}^{\frac{1}{4}} (2 {(x^{2} - x - 4)}^{\frac{1}{4}}) + {(x^{2} - x - 4)}^{\frac{1}{4}} {(x^{2} - x - 4)}^{\frac{1}{2}} + {(x^{2} - x - 4)}^{\frac{1}{4}} \cdot 4 - 2 (2 {(x^{2} - x - 4)}^{\frac{1}{4}}) - 2 {(x^{2} - x - 4)}^{\frac{1}{2}} - 2 \cdot 4 = 0$

Step 7.2

Simplify terms.

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Step 7.2.1

Simplify each term.

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Step 7.2.1.1

Rewrite using the commutative property of multiplication.

$2 {(x^{2} - x - 4)}^{\frac{1}{4}} {(x^{2} - x - 4)}^{\frac{1}{4}} + {(x^{2} - x - 4)}^{\frac{1}{4}} {(x^{2} - x - 4)}^{\frac{1}{2}} + {(x^{2} - x - 4)}^{\frac{1}{4}} \cdot 4 - 2 (2 {(x^{2} - x - 4)}^{\frac{1}{4}}) - 2 {(x^{2} - x - 4)}^{\frac{1}{2}} - 2 \cdot 4 = 0$

Step 7.2.1.2

Multiply ${(x^{2} - x - 4)}^{\frac{1}{4}}$ by ${(x^{2} - x - 4)}^{\frac{1}{4}}$ by adding the exponents.

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Step 7.2.1.2.1

Move ${(x^{2} - x - 4)}^{\frac{1}{4}}$ .

$2 ({(x^{2} - x - 4)}^{\frac{1}{4}} {(x^{2} - x - 4)}^{\frac{1}{4}}) + {(x^{2} - x - 4)}^{\frac{1}{4}} {(x^{2} - x - 4)}^{\frac{1}{2}} + {(x^{2} - x - 4)}^{\frac{1}{4}} \cdot 4 - 2 (2 {(x^{2} - x - 4)}^{\frac{1}{4}}) - 2 {(x^{2} - x - 4)}^{\frac{1}{2}} - 2 \cdot 4 = 0$

Step 7.2.1.2.2

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

$2 {(x^{2} - x - 4)}^{\frac{1}{4} + \frac{1}{4}} + {(x^{2} - x - 4)}^{\frac{1}{4}} {(x^{2} - x - 4)}^{\frac{1}{2}} + {(x^{2} - x - 4)}^{\frac{1}{4}} \cdot 4 - 2 (2 {(x^{2} - x - 4)}^{\frac{1}{4}}) - 2 {(x^{2} - x - 4)}^{\frac{1}{2}} - 2 \cdot 4 = 0$

Step 7.2.1.2.3

Combine the numerators over the common denominator.

$2 {(x^{2} - x - 4)}^{\frac{1 + 1}{4}} + {(x^{2} - x - 4)}^{\frac{1}{4}} {(x^{2} - x - 4)}^{\frac{1}{2}} + {(x^{2} - x - 4)}^{\frac{1}{4}} \cdot 4 - 2 (2 {(x^{2} - x - 4)}^{\frac{1}{4}}) - 2 {(x^{2} - x - 4)}^{\frac{1}{2}} - 2 \cdot 4 = 0$

Step 7.2.1.2.4

Add $1$ and $1$ .

$2 {(x^{2} - x - 4)}^{\frac{2}{4}} + {(x^{2} - x - 4)}^{\frac{1}{4}} {(x^{2} - x - 4)}^{\frac{1}{2}} + {(x^{2} - x - 4)}^{\frac{1}{4}} \cdot 4 - 2 (2 {(x^{2} - x - 4)}^{\frac{1}{4}}) - 2 {(x^{2} - x - 4)}^{\frac{1}{2}} - 2 \cdot 4 = 0$

Step 7.2.1.2.5

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

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Step 7.2.1.2.5.1

Factor $2$ out of $2$ .

$2 {(x^{2} - x - 4)}^{\frac{2 (1)}{4}} + {(x^{2} - x - 4)}^{\frac{1}{4}} {(x^{2} - x - 4)}^{\frac{1}{2}} + {(x^{2} - x - 4)}^{\frac{1}{4}} \cdot 4 - 2 (2 {(x^{2} - x - 4)}^{\frac{1}{4}}) - 2 {(x^{2} - x - 4)}^{\frac{1}{2}} - 2 \cdot 4 = 0$

Step 7.2.1.2.5.2

Cancel the common factors.

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Step 7.2.1.2.5.2.1

Factor $2$ out of $4$ .

$2 {(x^{2} - x - 4)}^{\frac{2 \cdot 1}{2 \cdot 2}} + {(x^{2} - x - 4)}^{\frac{1}{4}} {(x^{2} - x - 4)}^{\frac{1}{2}} + {(x^{2} - x - 4)}^{\frac{1}{4}} \cdot 4 - 2 (2 {(x^{2} - x - 4)}^{\frac{1}{4}}) - 2 {(x^{2} - x - 4)}^{\frac{1}{2}} - 2 \cdot 4 = 0$

Step 7.2.1.2.5.2.2

Cancel the common factor.

Step 7.2.1.2.5.2.3

Rewrite the expression.

$2 {(x^{2} - x - 4)}^{\frac{1}{2}} + {(x^{2} - x - 4)}^{\frac{1}{4}} {(x^{2} - x - 4)}^{\frac{1}{2}} + {(x^{2} - x - 4)}^{\frac{1}{4}} \cdot 4 - 2 (2 {(x^{2} - x - 4)}^{\frac{1}{4}}) - 2 {(x^{2} - x - 4)}^{\frac{1}{2}} - 2 \cdot 4 = 0$

Step 7.2.1.3

Multiply ${(x^{2} - x - 4)}^{\frac{1}{4}}$ by ${(x^{2} - x - 4)}^{\frac{1}{2}}$ by adding the exponents.

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Step 7.2.1.3.1

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

$2 {(x^{2} - x - 4)}^{\frac{1}{2}} + {(x^{2} - x - 4)}^{\frac{1}{4} + \frac{1}{2}} + {(x^{2} - x - 4)}^{\frac{1}{4}} \cdot 4 - 2 (2 {(x^{2} - x - 4)}^{\frac{1}{4}}) - 2 {(x^{2} - x - 4)}^{\frac{1}{2}} - 2 \cdot 4 = 0$

Step 7.2.1.3.2

To write $\frac{1}{2}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$2 {(x^{2} - x - 4)}^{\frac{1}{2}} + {(x^{2} - x - 4)}^{\frac{1}{4} + \frac{1}{2} \cdot \frac{2}{2}} + {(x^{2} - x - 4)}^{\frac{1}{4}} \cdot 4 - 2 (2 {(x^{2} - x - 4)}^{\frac{1}{4}}) - 2 {(x^{2} - x - 4)}^{\frac{1}{2}} - 2 \cdot 4 = 0$

Step 7.2.1.3.3

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

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Step 7.2.1.3.3.1

Multiply $\frac{1}{2}$ by $\frac{2}{2}$ .

$2 {(x^{2} - x - 4)}^{\frac{1}{2}} + {(x^{2} - x - 4)}^{\frac{1}{4} + \frac{2}{2 \cdot 2}} + {(x^{2} - x - 4)}^{\frac{1}{4}} \cdot 4 - 2 (2 {(x^{2} - x - 4)}^{\frac{1}{4}}) - 2 {(x^{2} - x - 4)}^{\frac{1}{2}} - 2 \cdot 4 = 0$

Step 7.2.1.3.3.2

Multiply $2$ by $2$ .

$2 {(x^{2} - x - 4)}^{\frac{1}{2}} + {(x^{2} - x - 4)}^{\frac{1}{4} + \frac{2}{4}} + {(x^{2} - x - 4)}^{\frac{1}{4}} \cdot 4 - 2 (2 {(x^{2} - x - 4)}^{\frac{1}{4}}) - 2 {(x^{2} - x - 4)}^{\frac{1}{2}} - 2 \cdot 4 = 0$

Step 7.2.1.3.4

Combine the numerators over the common denominator.

$2 {(x^{2} - x - 4)}^{\frac{1}{2}} + {(x^{2} - x - 4)}^{\frac{1 + 2}{4}} + {(x^{2} - x - 4)}^{\frac{1}{4}} \cdot 4 - 2 (2 {(x^{2} - x - 4)}^{\frac{1}{4}}) - 2 {(x^{2} - x - 4)}^{\frac{1}{2}} - 2 \cdot 4 = 0$

Step 7.2.1.3.5

Add $1$ and $2$ .

$2 {(x^{2} - x - 4)}^{\frac{1}{2}} + {(x^{2} - x - 4)}^{\frac{3}{4}} + {(x^{2} - x - 4)}^{\frac{1}{4}} \cdot 4 - 2 (2 {(x^{2} - x - 4)}^{\frac{1}{4}}) - 2 {(x^{2} - x - 4)}^{\frac{1}{2}} - 2 \cdot 4 = 0$

Step 7.2.1.4

Move $4$ to the left of ${(x^{2} - x - 4)}^{\frac{1}{4}}$ .

$2 {(x^{2} - x - 4)}^{\frac{1}{2}} + {(x^{2} - x - 4)}^{\frac{3}{4}} + 4 \cdot {(x^{2} - x - 4)}^{\frac{1}{4}} - 2 (2 {(x^{2} - x - 4)}^{\frac{1}{4}}) - 2 {(x^{2} - x - 4)}^{\frac{1}{2}} - 2 \cdot 4 = 0$

Step 7.2.1.5

Multiply $2$ by $- 2$ .

$2 {(x^{2} - x - 4)}^{\frac{1}{2}} + {(x^{2} - x - 4)}^{\frac{3}{4}} + 4 {(x^{2} - x - 4)}^{\frac{1}{4}} - 4 {(x^{2} - x - 4)}^{\frac{1}{4}} - 2 {(x^{2} - x - 4)}^{\frac{1}{2}} - 2 \cdot 4 = 0$

Step 7.2.1.6

Multiply $- 2$ by $4$ .

$2 {(x^{2} - x - 4)}^{\frac{1}{2}} + {(x^{2} - x - 4)}^{\frac{3}{4}} + 4 {(x^{2} - x - 4)}^{\frac{1}{4}} - 4 {(x^{2} - x - 4)}^{\frac{1}{4}} - 2 {(x^{2} - x - 4)}^{\frac{1}{2}} - 8 = 0$

Step 7.2.2

Combine the opposite terms in $2 {(x^{2} - x - 4)}^{\frac{1}{2}} + {(x^{2} - x - 4)}^{\frac{3}{4}} + 4 {(x^{2} - x - 4)}^{\frac{1}{4}} - 4 {(x^{2} - x - 4)}^{\frac{1}{4}} - 2 {(x^{2} - x - 4)}^{\frac{1}{2}} - 8$ .

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Step 7.2.2.1

Subtract $4 {(x^{2} - x - 4)}^{\frac{1}{4}}$ from $4 {(x^{2} - x - 4)}^{\frac{1}{4}}$ .

$2 {(x^{2} - x - 4)}^{\frac{1}{2}} + {(x^{2} - x - 4)}^{\frac{3}{4}} + 0 - 2 {(x^{2} - x - 4)}^{\frac{1}{2}} - 8 = 0$

Step 7.2.2.2

Add $2 {(x^{2} - x - 4)}^{\frac{1}{2}} + {(x^{2} - x - 4)}^{\frac{3}{4}}$ and $0$ .

$2 {(x^{2} - x - 4)}^{\frac{1}{2}} + {(x^{2} - x - 4)}^{\frac{3}{4}} - 2 {(x^{2} - x - 4)}^{\frac{1}{2}} - 8 = 0$

Step 7.2.2.3

Subtract $2 {(x^{2} - x - 4)}^{\frac{1}{2}}$ from $2 {(x^{2} - x - 4)}^{\frac{1}{2}}$ .

$0 + {(x^{2} - x - 4)}^{\frac{3}{4}} - 8 = 0$

Step 7.2.2.4

Add $0$ and ${(x^{2} - x - 4)}^{\frac{3}{4}}$ .

${(x^{2} - x - 4)}^{\frac{3}{4}} - 8 = 0$

Step 8

Add $8$ to both sides of the equation.

${(x^{2} - x - 4)}^{\frac{3}{4}} = 8$

Step 9

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

${({(x^{2} - x - 4)}^{\frac{3}{4}})}^{\frac{4}{3}} = 8^{\frac{4}{3}}$

Step 10

Simplify the exponent.

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Step 10.1

Simplify the left side.

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Step 10.1.1

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

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Step 10.1.1.1

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

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Step 10.1.1.1.1

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

${(x^{2} - x - 4)}^{\frac{3}{4} \cdot \frac{4}{3}} = 8^{\frac{4}{3}}$

Step 10.1.1.1.2

Cancel the common factor of $3$ .

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Step 10.1.1.1.2.1

Cancel the common factor.

${(x^{2} - x - 4)}^{\frac{3}{4} \cdot \frac{4}{3}} = 8^{\frac{4}{3}}$

Step 10.1.1.1.2.2

Rewrite the expression.

${(x^{2} - x - 4)}^{\frac{1}{4} \cdot 4} = 8^{\frac{4}{3}}$

Step 10.1.1.1.3

Cancel the common factor of $4$ .

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Step 10.1.1.1.3.1

Cancel the common factor.

${(x^{2} - x - 4)}^{\frac{1}{4} \cdot 4} = 8^{\frac{4}{3}}$

Step 10.1.1.1.3.2

Rewrite the expression.

${(x^{2} - x - 4)}^{1} = 8^{\frac{4}{3}}$

Step 10.1.1.2

Simplify.

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

Step 10.2

Simplify the right side.

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Step 10.2.1

Simplify $8^{\frac{4}{3}}$ .

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Step 10.2.1.1

Simplify the expression.

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Step 10.2.1.1.1

Rewrite $8$ as $2^{3}$ .

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

Step 10.2.1.1.2

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

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

Step 10.2.1.2

Cancel the common factor of $3$ .

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Step 10.2.1.2.1

Cancel the common factor.

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

Step 10.2.1.2.2

Rewrite the expression.

$x^{2} - x - 4 = 2^{4}$

Step 10.2.1.3

Raise $2$ to the power of $4$ .

$x^{2} - x - 4 = 16$

Step 11

Solve for $x$ .

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Step 11.1

Subtract $16$ from both sides of the equation.

$x^{2} - x - 4 - 16 = 0$

Step 11.2

Subtract $16$ from $- 4$ .

$x^{2} - x - 20 = 0$

Step 11.3

Factor $x^{2} - x - 20$ using the AC method.

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Step 11.3.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 20$ and whose sum is $- 1$ .

$- 5, 4$

Step 11.3.2

Write the factored form using these integers.

$(x - 5) (x + 4) = 0$

Step 11.4

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x - 5 = 0$

$x + 4 = 0$

Step 11.5

Set $x - 5$ equal to $0$ and solve for $x$ .

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Step 11.5.1

Set $x - 5$ equal to $0$ .

$x - 5 = 0$

Step 11.5.2

Add $5$ to both sides of the equation.

$x = 5$

Step 11.6

Set $x + 4$ equal to $0$ and solve for $x$ .

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Step 11.6.1

Set $x + 4$ equal to $0$ .

$x + 4 = 0$

Step 11.6.2

Subtract $4$ from both sides of the equation.

$x = - 4$

Step 11.7

The final solution is all the values that make $(x - 5) (x + 4) = 0$ true.

$x = 5, - 4$