Precalculus Examples

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

Step 1

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

$\sqrt{1 + (x^{3} - \frac{1}{4 x^{3}}) (x^{3} - \frac{1}{4 x^{3}})}$

Step 2

Expand $(x^{3} - \frac{1}{4 x^{3}}) (x^{3} - \frac{1}{4 x^{3}})$ using the FOIL Method.

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Step 2.1

Apply the distributive property.

$\sqrt{1 + x^{3} (x^{3} - \frac{1}{4 x^{3}}) - \frac{1}{4 x^{3}} (x^{3} - \frac{1}{4 x^{3}})}$

Step 2.2

Apply the distributive property.

$\sqrt{1 + x^{3} x^{3} + x^{3} (- \frac{1}{4 x^{3}}) - \frac{1}{4 x^{3}} (x^{3} - \frac{1}{4 x^{3}})}$

Step 2.3

Apply the distributive property.

$\sqrt{1 + x^{3} x^{3} + x^{3} (- \frac{1}{4 x^{3}}) - \frac{1}{4 x^{3}} x^{3} - \frac{1}{4 x^{3}} (- \frac{1}{4 x^{3}})}$

Step 3

Simplify and combine like terms.

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Step 3.1

Simplify each term.

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Step 3.1.1

Multiply $x^{3}$ by $x^{3}$ by adding the exponents.

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Step 3.1.1.1

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

$\sqrt{1 + x^{3 + 3} + x^{3} (- \frac{1}{4 x^{3}}) - \frac{1}{4 x^{3}} x^{3} - \frac{1}{4 x^{3}} (- \frac{1}{4 x^{3}})}$

Step 3.1.1.2

Add $3$ and $3$ .

$\sqrt{1 + x^{6} + x^{3} (- \frac{1}{4 x^{3}}) - \frac{1}{4 x^{3}} x^{3} - \frac{1}{4 x^{3}} (- \frac{1}{4 x^{3}})}$

Step 3.1.2

Rewrite using the commutative property of multiplication.

$\sqrt{1 + x^{6} - x^{3} \frac{1}{4 x^{3}} - \frac{1}{4 x^{3}} x^{3} - \frac{1}{4 x^{3}} (- \frac{1}{4 x^{3}})}$

Step 3.1.3

Cancel the common factor of $x^{3}$ .

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Step 3.1.3.1

Factor $x^{3}$ out of $- x^{3}$ .

$\sqrt{1 + x^{6} + x^{3} \cdot - 1 \frac{1}{4 x^{3}} - \frac{1}{4 x^{3}} x^{3} - \frac{1}{4 x^{3}} (- \frac{1}{4 x^{3}})}$

Step 3.1.3.2

Factor $x^{3}$ out of $4 x^{3}$ .

$\sqrt{1 + x^{6} + x^{3} \cdot - 1 \frac{1}{x^{3} \cdot 4} - \frac{1}{4 x^{3}} x^{3} - \frac{1}{4 x^{3}} (- \frac{1}{4 x^{3}})}$

Step 3.1.3.3

Cancel the common factor.

$\sqrt{1 + x^{6} + x^{3} \cdot - 1 \frac{1}{x^{3} \cdot 4} - \frac{1}{4 x^{3}} x^{3} - \frac{1}{4 x^{3}} (- \frac{1}{4 x^{3}})}$

Step 3.1.3.4

Rewrite the expression.

$\sqrt{1 + x^{6} - \frac{1}{4} - \frac{1}{4 x^{3}} x^{3} - \frac{1}{4 x^{3}} (- \frac{1}{4 x^{3}})}$

Step 3.1.4

Cancel the common factor of $x^{3}$ .

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Step 3.1.4.1

Move the leading negative in $- \frac{1}{4 x^{3}}$ into the numerator.

$\sqrt{1 + x^{6} - \frac{1}{4} + \frac{- 1}{4 x^{3}} x^{3} - \frac{1}{4 x^{3}} (- \frac{1}{4 x^{3}})}$

Step 3.1.4.2

Factor $x^{3}$ out of $4 x^{3}$ .

$\sqrt{1 + x^{6} - \frac{1}{4} + \frac{- 1}{x^{3} \cdot 4} x^{3} - \frac{1}{4 x^{3}} (- \frac{1}{4 x^{3}})}$

Step 3.1.4.3

Cancel the common factor.

$\sqrt{1 + x^{6} - \frac{1}{4} + \frac{- 1}{x^{3} \cdot 4} x^{3} - \frac{1}{4 x^{3}} (- \frac{1}{4 x^{3}})}$

Step 3.1.4.4

Rewrite the expression.

$\sqrt{1 + x^{6} - \frac{1}{4} + \frac{- 1}{4} - \frac{1}{4 x^{3}} (- \frac{1}{4 x^{3}})}$

Step 3.1.5

Move the negative in front of the fraction.

$\sqrt{1 + x^{6} - \frac{1}{4} - \frac{1}{4} - \frac{1}{4 x^{3}} (- \frac{1}{4 x^{3}})}$

Step 3.1.6

Multiply $- \frac{1}{4 x^{3}} (- \frac{1}{4 x^{3}})$ .

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Step 3.1.6.1

Multiply $- 1$ by $- 1$ .

$\sqrt{1 + x^{6} - \frac{1}{4} - \frac{1}{4} + 1 \frac{1}{4 x^{3}} \frac{1}{4 x^{3}}}$

Step 3.1.6.2

Multiply $\frac{1}{4 x^{3}}$ by $1$ .

$\sqrt{1 + x^{6} - \frac{1}{4} - \frac{1}{4} + \frac{1}{4 x^{3}} \cdot \frac{1}{4 x^{3}}}$

Step 3.1.6.3

Multiply $\frac{1}{4 x^{3}}$ by $\frac{1}{4 x^{3}}$ .

$\sqrt{1 + x^{6} - \frac{1}{4} - \frac{1}{4} + \frac{1}{4 x^{3} (4 x^{3})}}$

Step 3.1.6.4

Multiply $4$ by $4$ .

$\sqrt{1 + x^{6} - \frac{1}{4} - \frac{1}{4} + \frac{1}{16 x^{3} x^{3}}}$

Step 3.1.6.5

Multiply $x^{3}$ by $x^{3}$ by adding the exponents.

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Step 3.1.6.5.1

Move $x^{3}$ .

$\sqrt{1 + x^{6} - \frac{1}{4} - \frac{1}{4} + \frac{1}{16 (x^{3} x^{3})}}$

Step 3.1.6.5.2

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

$\sqrt{1 + x^{6} - \frac{1}{4} - \frac{1}{4} + \frac{1}{16 x^{3 + 3}}}$

Step 3.1.6.5.3

Add $3$ and $3$ .

$\sqrt{1 + x^{6} - \frac{1}{4} - \frac{1}{4} + \frac{1}{16 x^{6}}}$

Step 3.2

Subtract $\frac{1}{4}$ from $- \frac{1}{4}$ .

$\sqrt{1 + x^{6} - 2 (\frac{1}{4}) + \frac{1}{16 x^{6}}}$

Step 4

Simplify terms.

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Step 4.1

Simplify each term.

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Step 4.1.1

Cancel the common factor of $2$ .

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Step 4.1.1.1

Factor $2$ out of $- 2$ .

$\sqrt{1 + x^{6} + 2 (- 1) \frac{1}{4} + \frac{1}{16 x^{6}}}$

Step 4.1.1.2

Factor $2$ out of $4$ .

$\sqrt{1 + x^{6} + 2 \cdot - 1 \frac{1}{2 \cdot 2} + \frac{1}{16 x^{6}}}$

Step 4.1.1.3

Cancel the common factor.

$\sqrt{1 + x^{6} + 2 \cdot - 1 \frac{1}{2 \cdot 2} + \frac{1}{16 x^{6}}}$

Step 4.1.1.4

Rewrite the expression.

$\sqrt{1 + x^{6} - 1 (\frac{1}{2}) + \frac{1}{16 x^{6}}}$

Step 4.1.2

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

$\sqrt{1 + x^{6} - \frac{1}{2} + \frac{1}{16 x^{6}}}$

Step 4.2

Simplify the expression.

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Step 4.2.1

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

$\sqrt{x^{6} + \frac{2}{2} - \frac{1}{2} + \frac{1}{16 x^{6}}}$

Step 4.2.2

Combine the numerators over the common denominator.

$\sqrt{x^{6} + \frac{2 - 1}{2} + \frac{1}{16 x^{6}}}$

Step 4.2.3

Subtract $1$ from $2$ .

$\sqrt{x^{6} + \frac{1}{2} + \frac{1}{16 x^{6}}}$

Step 4.2.4

Reorder terms.

$\sqrt{x^{6} + \frac{1}{16 x^{6}} + \frac{1}{2}}$

Step 5

Factor using the perfect square rule.

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Step 5.1

Rearrange terms.

$\sqrt{x^{6} + \frac{1}{2} + \frac{1}{16 x^{6}}}$

Step 5.2

Rewrite $x^{6}$ as ${(x^{3})}^{2}$ .

$\sqrt{{(x^{3})}^{2} + \frac{1}{2} + \frac{1}{16 x^{6}}}$

Step 5.3

Rewrite $1$ as $1^{2}$ .

$\sqrt{{(x^{3})}^{2} + \frac{1}{2} + \frac{1^{2}}{16 x^{6}}}$

Step 5.4

Rewrite $16 x^{6}$ as ${(4 x^{3})}^{2}$ .

$\sqrt{{(x^{3})}^{2} + \frac{1}{2} + \frac{1^{2}}{{(4 x^{3})}^{2}}}$

Step 5.5

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

$\sqrt{{(x^{3})}^{2} + \frac{1}{2} + {(\frac{1}{4 x^{3}})}^{2}}$

Step 5.6

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

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

Step 5.7

Rewrite the polynomial.

$\sqrt{{(x^{3})}^{2} + 2 \cdot x^{3} \cdot \frac{1}{4 x^{3}} + {(\frac{1}{4 x^{3}})}^{2}}$

Step 5.8

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = x^{3}$ and $b = \frac{1}{4 x^{3}}$ .

$\sqrt{{(x^{3} + \frac{1}{4 x^{3}})}^{2}}$

Step 6

To write $x^{3}$ as a fraction with a common denominator, multiply by $\frac{4 x^{3}}{4 x^{3}}$ .

$\sqrt{{(x^{3} \cdot \frac{4 x^{3}}{4 x^{3}} + \frac{1}{4 x^{3}})}^{2}}$

Step 7

Simplify terms.

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Step 7.1

Combine $x^{3}$ and $\frac{4 x^{3}}{4 x^{3}}$ .

$\sqrt{{(\frac{x^{3} (4 x^{3})}{4 x^{3}} + \frac{1}{4 x^{3}})}^{2}}$

Step 7.2

Combine the numerators over the common denominator.

$\sqrt{{(\frac{x^{3} (4 x^{3}) + 1}{4 x^{3}})}^{2}}$

Step 8

Simplify the numerator.

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Step 8.1

Rewrite using the commutative property of multiplication.

$\sqrt{{(\frac{4 x^{3} x^{3} + 1}{4 x^{3}})}^{2}}$

Step 8.2

Multiply $x^{3}$ by $x^{3}$ by adding the exponents.

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Step 8.2.1

Move $x^{3}$ .

$\sqrt{{(\frac{4 (x^{3} x^{3}) + 1}{4 x^{3}})}^{2}}$

Step 8.2.2

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

$\sqrt{{(\frac{4 x^{3 + 3} + 1}{4 x^{3}})}^{2}}$

Step 8.2.3

Add $3$ and $3$ .

$\sqrt{{(\frac{4 x^{6} + 1}{4 x^{3}})}^{2}}$

Step 9

Apply basic rules of exponents.

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Step 9.1

Apply the product rule to $\frac{4 x^{6} + 1}{4 x^{3}}$ .

$\sqrt{\frac{{(4 x^{6} + 1)}^{2}}{{(4 x^{3})}^{2}}}$

Step 9.2

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

$\sqrt{\frac{{(4 x^{6} + 1)}^{2}}{4^{2} {(x^{3})}^{2}}}$

Step 10

Simplify the denominator.

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Step 10.1

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

$\sqrt{\frac{{(4 x^{6} + 1)}^{2}}{16 {(x^{3})}^{2}}}$

Step 10.2

Multiply the exponents in ${(x^{3})}^{2}$ .

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Step 10.2.1

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

$\sqrt{\frac{{(4 x^{6} + 1)}^{2}}{16 x^{3 \cdot 2}}}$

Step 10.2.2

Multiply $3$ by $2$ .

$\sqrt{\frac{{(4 x^{6} + 1)}^{2}}{16 x^{6}}}$

Step 11

Rewrite $16 x^{6}$ as ${(4 x^{3})}^{2}$ .

$\sqrt{\frac{{(4 x^{6} + 1)}^{2}}{{(4 x^{3})}^{2}}}$

Step 12

Rewrite $\frac{{(4 x^{6} + 1)}^{2}}{{(4 x^{3})}^{2}}$ as ${(\frac{4 x^{6} + 1}{4 x^{3}})}^{2}$ .

$\sqrt{{(\frac{4 x^{6} + 1}{4 x^{3}})}^{2}}$

Step 13

Pull terms out from under the radical, assuming positive real numbers.

$\frac{4 x^{6} + 1}{4 x^{3}}$