Algebra Examples

$x^{\frac{3}{4}} = 729$

Step 1

Subtract $729$ from both sides of the equation.

$x^{\frac{3}{4}} - 729 = 0$

Step 2

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

${(x^{\frac{1}{4}})}^{3} - 729 = 0$

Step 3

Rewrite $729$ as $9^{3}$ .

${(x^{\frac{1}{4}})}^{3} - 9^{3} = 0$

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 $a = x^{\frac{1}{4}}$ and $b = 9$ .

$(x^{\frac{1}{4}} - 9) ({(x^{\frac{1}{4}})}^{2} + x^{\frac{1}{4}} \cdot 9 + 9^{2}) = 0$

Step 5

Simplify.

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Step 5.1

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

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Step 5.1.1

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

$(x^{\frac{1}{4}} - 9) (x^{\frac{1}{4} \cdot 2} + x^{\frac{1}{4}} \cdot 9 + 9^{2}) = 0$

Step 5.1.2

Cancel the common factor of $2$ .

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Step 5.1.2.1

Factor $2$ out of $4$ .

$(x^{\frac{1}{4}} - 9) (x^{\frac{1}{2 (2)} \cdot 2} + x^{\frac{1}{4}} \cdot 9 + 9^{2}) = 0$

Step 5.1.2.2

Cancel the common factor.

$(x^{\frac{1}{4}} - 9) (x^{\frac{1}{2 \cdot 2} \cdot 2} + x^{\frac{1}{4}} \cdot 9 + 9^{2}) = 0$

Step 5.1.2.3

Rewrite the expression.

$(x^{\frac{1}{4}} - 9) (x^{\frac{1}{2}} + x^{\frac{1}{4}} \cdot 9 + 9^{2}) = 0$

Step 5.2

Move $9$ to the left of $x^{\frac{1}{4}}$ .

$(x^{\frac{1}{4}} - 9) (x^{\frac{1}{2}} + 9 x^{\frac{1}{4}} + 9^{2}) = 0$

Step 5.3

Raise $9$ to the power of $2$ .

$(x^{\frac{1}{4}} - 9) (x^{\frac{1}{2}} + 9 x^{\frac{1}{4}} + 81) = 0$

Step 5.4

Reorder terms.

$(x^{\frac{1}{4}} - 9) (9 x^{\frac{1}{4}} + x^{\frac{1}{2}} + 81) = 0$

Step 6

Simplify $(x^{\frac{1}{4}} - 9) (9 x^{\frac{1}{4}} + x^{\frac{1}{2}} + 81)$ .

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Step 6.1

Expand $(x^{\frac{1}{4}} - 9) (9 x^{\frac{1}{4}} + x^{\frac{1}{2}} + 81)$ by multiplying each term in the first expression by each term in the second expression.

$x^{\frac{1}{4}} (9 x^{\frac{1}{4}}) + x^{\frac{1}{4}} x^{\frac{1}{2}} + x^{\frac{1}{4}} \cdot 81 - 9 (9 x^{\frac{1}{4}}) - 9 x^{\frac{1}{2}} - 9 \cdot 81 = 0$

Step 6.2

Simplify terms.

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Step 6.2.1

Simplify each term.

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Step 6.2.1.1

Rewrite using the commutative property of multiplication.

$9 x^{\frac{1}{4}} x^{\frac{1}{4}} + x^{\frac{1}{4}} x^{\frac{1}{2}} + x^{\frac{1}{4}} \cdot 81 - 9 (9 x^{\frac{1}{4}}) - 9 x^{\frac{1}{2}} - 9 \cdot 81 = 0$

Step 6.2.1.2

Multiply $x^{\frac{1}{4}}$ by $x^{\frac{1}{4}}$ by adding the exponents.

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Step 6.2.1.2.1

Move $x^{\frac{1}{4}}$ .

$9 (x^{\frac{1}{4}} x^{\frac{1}{4}}) + x^{\frac{1}{4}} x^{\frac{1}{2}} + x^{\frac{1}{4}} \cdot 81 - 9 (9 x^{\frac{1}{4}}) - 9 x^{\frac{1}{2}} - 9 \cdot 81 = 0$

Step 6.2.1.2.2

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

$9 x^{\frac{1}{4} + \frac{1}{4}} + x^{\frac{1}{4}} x^{\frac{1}{2}} + x^{\frac{1}{4}} \cdot 81 - 9 (9 x^{\frac{1}{4}}) - 9 x^{\frac{1}{2}} - 9 \cdot 81 = 0$

Step 6.2.1.2.3

Combine the numerators over the common denominator.

$9 x^{\frac{1 + 1}{4}} + x^{\frac{1}{4}} x^{\frac{1}{2}} + x^{\frac{1}{4}} \cdot 81 - 9 (9 x^{\frac{1}{4}}) - 9 x^{\frac{1}{2}} - 9 \cdot 81 = 0$

Step 6.2.1.2.4

Add $1$ and $1$ .

$9 x^{\frac{2}{4}} + x^{\frac{1}{4}} x^{\frac{1}{2}} + x^{\frac{1}{4}} \cdot 81 - 9 (9 x^{\frac{1}{4}}) - 9 x^{\frac{1}{2}} - 9 \cdot 81 = 0$

Step 6.2.1.2.5

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

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Step 6.2.1.2.5.1

Factor $2$ out of $2$ .

$9 x^{\frac{2 (1)}{4}} + x^{\frac{1}{4}} x^{\frac{1}{2}} + x^{\frac{1}{4}} \cdot 81 - 9 (9 x^{\frac{1}{4}}) - 9 x^{\frac{1}{2}} - 9 \cdot 81 = 0$

Step 6.2.1.2.5.2

Cancel the common factors.

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Step 6.2.1.2.5.2.1

Factor $2$ out of $4$ .

$9 x^{\frac{2 \cdot 1}{2 \cdot 2}} + x^{\frac{1}{4}} x^{\frac{1}{2}} + x^{\frac{1}{4}} \cdot 81 - 9 (9 x^{\frac{1}{4}}) - 9 x^{\frac{1}{2}} - 9 \cdot 81 = 0$

Step 6.2.1.2.5.2.2

Cancel the common factor.

$9 x^{\frac{2 \cdot 1}{2 \cdot 2}} + x^{\frac{1}{4}} x^{\frac{1}{2}} + x^{\frac{1}{4}} \cdot 81 - 9 (9 x^{\frac{1}{4}}) - 9 x^{\frac{1}{2}} - 9 \cdot 81 = 0$

Step 6.2.1.2.5.2.3

Rewrite the expression.

$9 x^{\frac{1}{2}} + x^{\frac{1}{4}} x^{\frac{1}{2}} + x^{\frac{1}{4}} \cdot 81 - 9 (9 x^{\frac{1}{4}}) - 9 x^{\frac{1}{2}} - 9 \cdot 81 = 0$

Step 6.2.1.3

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

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Step 6.2.1.3.1

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

$9 x^{\frac{1}{2}} + x^{\frac{1}{4} + \frac{1}{2}} + x^{\frac{1}{4}} \cdot 81 - 9 (9 x^{\frac{1}{4}}) - 9 x^{\frac{1}{2}} - 9 \cdot 81 = 0$

Step 6.2.1.3.2

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

$9 x^{\frac{1}{2}} + x^{\frac{1}{4} + \frac{1}{2} \cdot \frac{2}{2}} + x^{\frac{1}{4}} \cdot 81 - 9 (9 x^{\frac{1}{4}}) - 9 x^{\frac{1}{2}} - 9 \cdot 81 = 0$

Step 6.2.1.3.3

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

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Step 6.2.1.3.3.1

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

$9 x^{\frac{1}{2}} + x^{\frac{1}{4} + \frac{2}{2 \cdot 2}} + x^{\frac{1}{4}} \cdot 81 - 9 (9 x^{\frac{1}{4}}) - 9 x^{\frac{1}{2}} - 9 \cdot 81 = 0$

Step 6.2.1.3.3.2

Multiply $2$ by $2$ .

$9 x^{\frac{1}{2}} + x^{\frac{1}{4} + \frac{2}{4}} + x^{\frac{1}{4}} \cdot 81 - 9 (9 x^{\frac{1}{4}}) - 9 x^{\frac{1}{2}} - 9 \cdot 81 = 0$

Step 6.2.1.3.4

Combine the numerators over the common denominator.

$9 x^{\frac{1}{2}} + x^{\frac{1 + 2}{4}} + x^{\frac{1}{4}} \cdot 81 - 9 (9 x^{\frac{1}{4}}) - 9 x^{\frac{1}{2}} - 9 \cdot 81 = 0$

Step 6.2.1.3.5

Add $1$ and $2$ .

$9 x^{\frac{1}{2}} + x^{\frac{3}{4}} + x^{\frac{1}{4}} \cdot 81 - 9 (9 x^{\frac{1}{4}}) - 9 x^{\frac{1}{2}} - 9 \cdot 81 = 0$

Step 6.2.1.4

Move $81$ to the left of $x^{\frac{1}{4}}$ .

$9 x^{\frac{1}{2}} + x^{\frac{3}{4}} + 81 \cdot x^{\frac{1}{4}} - 9 (9 x^{\frac{1}{4}}) - 9 x^{\frac{1}{2}} - 9 \cdot 81 = 0$

Step 6.2.1.5

Multiply $9$ by $- 9$ .

$9 x^{\frac{1}{2}} + x^{\frac{3}{4}} + 81 x^{\frac{1}{4}} - 81 x^{\frac{1}{4}} - 9 x^{\frac{1}{2}} - 9 \cdot 81 = 0$

Step 6.2.1.6

Multiply $- 9$ by $81$ .

$9 x^{\frac{1}{2}} + x^{\frac{3}{4}} + 81 x^{\frac{1}{4}} - 81 x^{\frac{1}{4}} - 9 x^{\frac{1}{2}} - 729 = 0$

Step 6.2.2

Combine the opposite terms in $9 x^{\frac{1}{2}} + x^{\frac{3}{4}} + 81 x^{\frac{1}{4}} - 81 x^{\frac{1}{4}} - 9 x^{\frac{1}{2}} - 729$ .

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Step 6.2.2.1

Subtract $81 x^{\frac{1}{4}}$ from $81 x^{\frac{1}{4}}$ .

$9 x^{\frac{1}{2}} + x^{\frac{3}{4}} + 0 - 9 x^{\frac{1}{2}} - 729 = 0$

Step 6.2.2.2

Add $9 x^{\frac{1}{2}} + x^{\frac{3}{4}}$ and $0$ .

$9 x^{\frac{1}{2}} + x^{\frac{3}{4}} - 9 x^{\frac{1}{2}} - 729 = 0$

Step 6.2.2.3

Subtract $9 x^{\frac{1}{2}}$ from $9 x^{\frac{1}{2}}$ .

$x^{\frac{3}{4}} + 0 - 729 = 0$

Step 6.2.2.4

Add $x^{\frac{3}{4}}$ and $0$ .

$x^{\frac{3}{4}} - 729 = 0$

Step 7

Add $729$ to both sides of the equation.

$x^{\frac{3}{4}} = 729$

Step 8

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

${(x^{\frac{3}{4}})}^{\frac{4}{3}} = 729^{\frac{4}{3}}$

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

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Step 9.1.1.1

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

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Step 9.1.1.1.1

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

$x^{\frac{3}{4} \cdot \frac{4}{3}} = 729^{\frac{4}{3}}$

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.

$x^{\frac{3}{4} \cdot \frac{4}{3}} = 729^{\frac{4}{3}}$

Step 9.1.1.1.2.2

Rewrite the expression.

$x^{\frac{1}{4} \cdot 4} = 729^{\frac{4}{3}}$

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.

$x^{\frac{1}{4} \cdot 4} = 729^{\frac{4}{3}}$

Step 9.1.1.1.3.2

Rewrite the expression.

$x^{1} = 729^{\frac{4}{3}}$

Step 9.1.1.2

Simplify.

$x = 729^{\frac{4}{3}}$

Step 9.2

Simplify the right side.

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Step 9.2.1

Simplify $729^{\frac{4}{3}}$ .

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Step 9.2.1.1

Simplify the expression.

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Step 9.2.1.1.1

Rewrite $729$ as $9^{3}$ .

$x = {(9^{3})}^{\frac{4}{3}}$

Step 9.2.1.1.2

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

$x = 9^{3 (\frac{4}{3})}$

Step 9.2.1.2

Cancel the common factor of $3$ .

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Step 9.2.1.2.1

Cancel the common factor.

$x = 9^{3 (\frac{4}{3})}$

Step 9.2.1.2.2

Rewrite the expression.

$x = 9^{4}$

Step 9.2.1.3

Raise $9$ to the power of $4$ .

$x = 6561$