Algebra Examples

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

Step 1

Subtract $16$ from both sides of the equation.

$2 x^{\frac{3}{4}} - 16 = 0$

Step 2

Factor $2$ out of $2 x^{\frac{3}{4}} - 16$ .

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Step 2.1

Factor $2$ out of $2 x^{\frac{3}{4}}$ .

$2 (x^{\frac{3}{4}}) - 16 = 0$

Step 2.2

Factor $2$ out of $- 16$ .

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

Step 2.3

Factor $2$ out of $2 x^{\frac{3}{4}} + 2 \cdot - 8$ .

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

Step 3

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

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

Step 4

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

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

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

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

Step 6

Factor.

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Step 6.1

Simplify.

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Step 6.1.1

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

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Step 6.1.1.1

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

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

Step 6.1.1.2

Cancel the common factor of $2$ .

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Step 6.1.1.2.1

Factor $2$ out of $4$ .

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

Step 6.1.1.2.2

Cancel the common factor.

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

Step 6.1.1.2.3

Rewrite the expression.

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

Step 6.1.2

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

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

Step 6.1.3

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

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

Step 6.1.4

Reorder terms.

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

Step 6.2

Remove unnecessary parentheses.

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

Step 7

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

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Step 7.1

Simplify by multiplying through.

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Step 7.1.1

Apply the distributive property.

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

Step 7.1.2

Multiply $2$ by $- 2$ .

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

Step 7.2

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

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

Step 7.3

Simplify terms.

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Step 7.3.1

Simplify each term.

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Step 7.3.1.1

Rewrite using the commutative property of multiplication.

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

Step 7.3.1.2

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

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Step 7.3.1.2.1

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

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

Step 7.3.1.2.2

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

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

Step 7.3.1.2.3

Combine the numerators over the common denominator.

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

Step 7.3.1.2.4

Add $1$ and $1$ .

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

Step 7.3.1.2.5

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

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Step 7.3.1.2.5.1

Factor $2$ out of $2$ .

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

Step 7.3.1.2.5.2

Cancel the common factors.

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Step 7.3.1.2.5.2.1

Factor $2$ out of $4$ .

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

Step 7.3.1.2.5.2.2

Cancel the common factor.

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

Step 7.3.1.2.5.2.3

Rewrite the expression.

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

Step 7.3.1.3

Multiply $2$ by $2$ .

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

Step 7.3.1.4

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

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Step 7.3.1.4.1

Move $x^{\frac{1}{2}}$ .

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

Step 7.3.1.4.2

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

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

Step 7.3.1.4.3

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

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

Step 7.3.1.4.4

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

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Step 7.3.1.4.4.1

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

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

Step 7.3.1.4.4.2

Multiply $2$ by $2$ .

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

Step 7.3.1.4.5

Combine the numerators over the common denominator.

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

Step 7.3.1.4.6

Add $2$ and $1$ .

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

Step 7.3.1.5

Multiply $4$ by $2$ .

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

Step 7.3.1.6

Multiply $2$ by $- 4$ .

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

Step 7.3.1.7

Multiply $- 4$ by $4$ .

$4 x^{\frac{1}{2}} + 2 x^{\frac{3}{4}} + 8 x^{\frac{1}{4}} - 8 x^{\frac{1}{4}} - 4 x^{\frac{1}{2}} - 16 = 0$

Step 7.3.2

Combine the opposite terms in $4 x^{\frac{1}{2}} + 2 x^{\frac{3}{4}} + 8 x^{\frac{1}{4}} - 8 x^{\frac{1}{4}} - 4 x^{\frac{1}{2}} - 16$ .

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Step 7.3.2.1

Subtract $8 x^{\frac{1}{4}}$ from $8 x^{\frac{1}{4}}$ .

$4 x^{\frac{1}{2}} + 2 x^{\frac{3}{4}} + 0 - 4 x^{\frac{1}{2}} - 16 = 0$

Step 7.3.2.2

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

$4 x^{\frac{1}{2}} + 2 x^{\frac{3}{4}} - 4 x^{\frac{1}{2}} - 16 = 0$

Step 7.3.2.3

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

$2 x^{\frac{3}{4}} + 0 - 16 = 0$

Step 7.3.2.4

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

$2 x^{\frac{3}{4}} - 16 = 0$

Step 8

Add $16$ to both sides of the equation.

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

Step 9

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

${(2 x^{\frac{3}{4}})}^{\frac{4}{3}} = 16^{\frac{4}{3}}$

Step 10

Simplify the left side.

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Step 10.1

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

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Step 10.1.1

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

$2^{\frac{4}{3}} {(x^{\frac{3}{4}})}^{\frac{4}{3}} = 16^{\frac{4}{3}}$

Step 10.1.2

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

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Step 10.1.2.1

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

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

Step 10.1.2.2

Cancel the common factor of $3$ .

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Step 10.1.2.2.1

Cancel the common factor.

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

Step 10.1.2.2.2

Rewrite the expression.

$2^{\frac{4}{3}} x^{\frac{1}{4} \cdot 4} = 16^{\frac{4}{3}}$

Step 10.1.2.3

Cancel the common factor of $4$ .

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Step 10.1.2.3.1

Cancel the common factor.

$2^{\frac{4}{3}} x^{\frac{1}{4} \cdot 4} = 16^{\frac{4}{3}}$

Step 10.1.2.3.2

Rewrite the expression.

$2^{\frac{4}{3}} x^{1} = 16^{\frac{4}{3}}$

Step 10.1.3

Simplify.

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

Step 10.1.4

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

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

Step 11

Divide each term in $x \cdot 2^{\frac{4}{3}} = 16^{\frac{4}{3}}$ by $2^{\frac{4}{3}}$ and simplify.

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Step 11.1

Divide each term in $x \cdot 2^{\frac{4}{3}} = 16^{\frac{4}{3}}$ by $2^{\frac{4}{3}}$ .

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

Step 11.2

Simplify the left side.

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Step 11.2.1

Cancel the common factor.

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

Step 11.2.2

Divide $x$ by $1$ .

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

Step 11.3

Simplify the right side.

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Step 11.3.1

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

$x = {(\frac{16}{2})}^{\frac{4}{3}}$

Step 11.3.2

Simplify the expression.

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Step 11.3.2.1

Divide $16$ by $2$ .

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

Step 11.3.2.2

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

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

Step 11.3.2.3

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

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

Step 11.3.3

Cancel the common factor of $3$ .

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Step 11.3.3.1

Cancel the common factor.

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

Step 11.3.3.2

Rewrite the expression.

$x = 2^{4}$

Step 11.3.4

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

$x = 16$