Algebra Examples

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

Step 1

Subtract $1$ from both sides of the equation.

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

Step 2

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

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

Step 3

Rewrite $1$ as $1^{3}$ .

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

Step 4

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 $a = {(x - 2)}^{\frac{1}{4}}$ and $b = 1$ .

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

Step 5

Simplify.

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Step 5.1

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

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Step 5.1.1

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

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

Step 5.1.2

Cancel the common factor of $2$ .

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Step 5.1.2.1

Factor $2$ out of $4$ .

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

Step 5.1.2.2

Cancel the common factor.

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

Step 5.1.2.3

Rewrite the expression.

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

Step 5.2

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

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

Step 5.3

One to any power is one.

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

Step 6

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

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Step 6.1

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

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

Step 6.2

Simplify terms.

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Step 6.2.1

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

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Step 6.2.1.1

Reorder the factors in the terms ${(x - 2)}^{\frac{1}{4}} \cdot 1$ and $- 1 {(x - 2)}^{\frac{1}{4}}$ .

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

Step 6.2.1.2

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

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

Step 6.2.1.3

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

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

Step 6.2.2

Simplify each term.

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Step 6.2.2.1

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

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Step 6.2.2.1.1

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

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

Step 6.2.2.1.2

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

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

Step 6.2.2.1.3

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

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Step 6.2.2.1.3.1

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

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

Step 6.2.2.1.3.2

Multiply $2$ by $2$ .

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

Step 6.2.2.1.4

Combine the numerators over the common denominator.

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

Step 6.2.2.1.5

Add $1$ and $2$ .

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

Step 6.2.2.2

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

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Step 6.2.2.2.1

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

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

Step 6.2.2.2.2

Combine the numerators over the common denominator.

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

Step 6.2.2.2.3

Add $1$ and $1$ .

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

Step 6.2.2.2.4

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

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Step 6.2.2.2.4.1

Factor $2$ out of $2$ .

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

Step 6.2.2.2.4.2

Cancel the common factors.

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Step 6.2.2.2.4.2.1

Factor $2$ out of $4$ .

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

Step 6.2.2.2.4.2.2

Cancel the common factor.

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

Step 6.2.2.2.4.2.3

Rewrite the expression.

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

Step 6.2.2.3

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

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

Step 6.2.2.4

Multiply $- 1$ by $1$ .

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

Step 6.2.3

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

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Step 6.2.3.1

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

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

Step 6.2.3.2

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

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

Step 7

Add $1$ to both sides of the equation.

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

Step 8

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

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

Step 9

Simplify the exponent.

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Step 9.1

Simplify the left side.

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Step 9.1.1

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

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Step 9.1.1.1

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

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Step 9.1.1.1.1

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

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

Step 9.1.1.1.2

Cancel the common factor of $3$ .

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Step 9.1.1.1.2.1

Cancel the common factor.

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

Step 9.1.1.1.2.2

Rewrite the expression.

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

Step 9.1.1.1.3

Cancel the common factor of $4$ .

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Step 9.1.1.1.3.1

Cancel the common factor.

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

Step 9.1.1.1.3.2

Rewrite the expression.

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

Step 9.1.1.2

Simplify.

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

Step 9.2

Simplify the right side.

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Step 9.2.1

One to any power is one.

$x - 2 = 1$

Step 10

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

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Step 10.1

Add $2$ to both sides of the equation.

$x = 1 + 2$

Step 10.2

Add $1$ and $2$ .

$x = 3$