Pre-Algebra Examples

$\frac{- 7 \sqrt{x} - 5}{\sqrt{x} + 9}$

Step 1

Factor $- 1$ out of $- 7 \sqrt{x} - 5$ .

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Step 1.1

Factor $- 1$ out of $- 7 \sqrt{x}$ .

$\frac{- (7 \sqrt{x}) - 5}{\sqrt{x} + 9}$

Step 1.2

Rewrite $- 5$ as $- 1 (5)$ .

$\frac{- (7 \sqrt{x}) - 1 (5)}{\sqrt{x} + 9}$

Step 1.3

Factor $- 1$ out of $- (7 \sqrt{x}) - 1 (5)$ .

$\frac{- (7 \sqrt{x} + 5)}{\sqrt{x} + 9}$

Step 2

Move the negative in front of the fraction.

$- \frac{7 \sqrt{x} + 5}{\sqrt{x} + 9}$

Step 3

Multiply $\frac{7 \sqrt{x} + 5}{\sqrt{x} + 9}$ by $\frac{\sqrt{x} - 9}{\sqrt{x} - 9}$ .

$- (\frac{7 \sqrt{x} + 5}{\sqrt{x} + 9} \cdot \frac{\sqrt{x} - 9}{\sqrt{x} - 9})$

Step 4

Multiply $\frac{7 \sqrt{x} + 5}{\sqrt{x} + 9}$ by $\frac{\sqrt{x} - 9}{\sqrt{x} - 9}$ .

$- \frac{(7 \sqrt{x} + 5) (\sqrt{x} - 9)}{(\sqrt{x} + 9) (\sqrt{x} - 9)}$

Step 5

Expand the denominator using the FOIL method.

$- \frac{(7 \sqrt{x} + 5) (\sqrt{x} - 9)}{{\sqrt{x}}^{2} + \sqrt{x} \cdot - 9 + 9 \sqrt{x} - 81}$

Step 6

Simplify.

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Step 6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x}$ as $x^{\frac{1}{2}}$ .

$- \frac{(7 \sqrt{x} + 5) (\sqrt{x} - 9)}{{(x^{\frac{1}{2}})}^{2} - 81}$

Step 6.2

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

$- \frac{(7 \sqrt{x} + 5) (\sqrt{x} - 9)}{x^{\frac{1}{2} \cdot 2} - 81}$

Step 6.3

Combine $\frac{1}{2}$ and $2$ .

$- \frac{(7 \sqrt{x} + 5) (\sqrt{x} - 9)}{x^{\frac{2}{2}} - 81}$

Step 6.4

Cancel the common factor of $2$ .

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Step 6.4.1

Cancel the common factor.

$- \frac{(7 \sqrt{x} + 5) (\sqrt{x} - 9)}{x^{\frac{2}{2}} - 81}$

Step 6.4.2

Rewrite the expression.

$- \frac{(7 \sqrt{x} + 5) (\sqrt{x} - 9)}{x^{1} - 81}$

Step 6.5

Simplify.

$- \frac{(7 \sqrt{x} + 5) (\sqrt{x} - 9)}{x - 81}$

Step 7

Expand $(7 \sqrt{x} + 5) (\sqrt{x} - 9)$ using the FOIL Method.

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Step 7.1

Apply the distributive property.

$- \frac{7 \sqrt{x} (\sqrt{x} - 9) + 5 (\sqrt{x} - 9)}{x - 81}$

Step 7.2

Apply the distributive property.

$- \frac{7 \sqrt{x} \sqrt{x} + 7 \sqrt{x} \cdot - 9 + 5 (\sqrt{x} - 9)}{x - 81}$

Step 7.3

Apply the distributive property.

$- \frac{7 \sqrt{x} \sqrt{x} + 7 \sqrt{x} \cdot - 9 + 5 \sqrt{x} + 5 \cdot - 9}{x - 81}$

Step 8

Simplify and combine like terms.

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Step 8.1

Simplify each term.

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Step 8.1.1

Multiply $7 \sqrt{x} \sqrt{x}$ .

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Step 8.1.1.1

Raise $\sqrt{x}$ to the power of $1$ .

$- \frac{7 ({\sqrt{x}}^{1} \sqrt{x}) + 7 \sqrt{x} \cdot - 9 + 5 \sqrt{x} + 5 \cdot - 9}{x - 81}$

Step 8.1.1.2

Raise $\sqrt{x}$ to the power of $1$ .

$- \frac{7 ({\sqrt{x}}^{1} {\sqrt{x}}^{1}) + 7 \sqrt{x} \cdot - 9 + 5 \sqrt{x} + 5 \cdot - 9}{x - 81}$

Step 8.1.1.3

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

$- \frac{7 {\sqrt{x}}^{1 + 1} + 7 \sqrt{x} \cdot - 9 + 5 \sqrt{x} + 5 \cdot - 9}{x - 81}$

Step 8.1.1.4

Add $1$ and $1$ .

$- \frac{7 {\sqrt{x}}^{2} + 7 \sqrt{x} \cdot - 9 + 5 \sqrt{x} + 5 \cdot - 9}{x - 81}$

Step 8.1.2

Rewrite ${\sqrt{x}}^{2}$ as $x$ .

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Step 8.1.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x}$ as $x^{\frac{1}{2}}$ .

$- \frac{7 {(x^{\frac{1}{2}})}^{2} + 7 \sqrt{x} \cdot - 9 + 5 \sqrt{x} + 5 \cdot - 9}{x - 81}$

Step 8.1.2.2

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

$- \frac{7 x^{\frac{1}{2} \cdot 2} + 7 \sqrt{x} \cdot - 9 + 5 \sqrt{x} + 5 \cdot - 9}{x - 81}$

Step 8.1.2.3

Combine $\frac{1}{2}$ and $2$ .

$- \frac{7 x^{\frac{2}{2}} + 7 \sqrt{x} \cdot - 9 + 5 \sqrt{x} + 5 \cdot - 9}{x - 81}$

Step 8.1.2.4

Cancel the common factor of $2$ .

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Step 8.1.2.4.1

Cancel the common factor.

$- \frac{7 x^{\frac{2}{2}} + 7 \sqrt{x} \cdot - 9 + 5 \sqrt{x} + 5 \cdot - 9}{x - 81}$

Step 8.1.2.4.2

Rewrite the expression.

$- \frac{7 x^{1} + 7 \sqrt{x} \cdot - 9 + 5 \sqrt{x} + 5 \cdot - 9}{x - 81}$

Step 8.1.2.5

Simplify.

$- \frac{7 x + 7 \sqrt{x} \cdot - 9 + 5 \sqrt{x} + 5 \cdot - 9}{x - 81}$

Step 8.1.3

Multiply $- 9$ by $7$ .

$- \frac{7 x - 63 \sqrt{x} + 5 \sqrt{x} + 5 \cdot - 9}{x - 81}$

Step 8.1.4

Multiply $5$ by $- 9$ .

$- \frac{7 x - 63 \sqrt{x} + 5 \sqrt{x} - 45}{x - 81}$

Step 8.2

Add $- 63 \sqrt{x}$ and $5 \sqrt{x}$ .

$- \frac{7 x - 58 \sqrt{x} - 45}{x - 81}$

Step 9

Simplify the numerator.

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Step 9.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x}$ as $x^{\frac{1}{2}}$ .

$- \frac{7 x - 58 x^{\frac{1}{2}} - 45}{x - 81}$

Step 9.2

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

$- \frac{7 {(x^{\frac{1}{2}})}^{2} - 58 x^{\frac{1}{2}} - 45}{x - 81}$

Step 9.3

Let $u = x^{\frac{1}{2}}$ . Substitute $u$ for all occurrences of $x^{\frac{1}{2}}$ .

$- \frac{7 u^{2} - 58 u - 45}{x - 81}$

Step 9.4

Factor by grouping.

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Step 9.4.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 7 \cdot - 45 = - 315$ and whose sum is $b = - 58$ .

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Step 9.4.1.1

Factor $- 58$ out of $- 58 u$ .

$- \frac{7 u^{2} - 58 (u) - 45}{x - 81}$

Step 9.4.1.2

Rewrite $- 58$ as $5$ plus $- 63$

$- \frac{7 u^{2} + (5 - 63) u - 45}{x - 81}$

Step 9.4.1.3

Apply the distributive property.

$- \frac{7 u^{2} + 5 u - 63 u - 45}{x - 81}$

Step 9.4.2

Factor out the greatest common factor from each group.

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Step 9.4.2.1

Group the first two terms and the last two terms.

$- \frac{(7 u^{2} + 5 u) - 63 u - 45}{x - 81}$

Step 9.4.2.2

Factor out the greatest common factor (GCF) from each group.

$- \frac{u (7 u + 5) - 9 (7 u + 5)}{x - 81}$

Step 9.4.3

Factor the polynomial by factoring out the greatest common factor, $7 u + 5$ .

$- \frac{(7 u + 5) (u - 9)}{x - 81}$

Step 9.5

Replace all occurrences of $u$ with $x^{\frac{1}{2}}$ .

$- \frac{(7 x^{\frac{1}{2}} + 5) (x^{\frac{1}{2}} - 9)}{x - 81}$

Step 10

Expand $(7 x^{\frac{1}{2}} + 5) (x^{\frac{1}{2}} - 9)$ using the FOIL Method.

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Step 10.1

Apply the distributive property.

$- \frac{7 x^{\frac{1}{2}} (x^{\frac{1}{2}} - 9) + 5 (x^{\frac{1}{2}} - 9)}{x - 81}$

Step 10.2

Apply the distributive property.

$- \frac{7 x^{\frac{1}{2}} x^{\frac{1}{2}} + 7 x^{\frac{1}{2}} \cdot - 9 + 5 (x^{\frac{1}{2}} - 9)}{x - 81}$

Step 10.3

Apply the distributive property.

$- \frac{7 x^{\frac{1}{2}} x^{\frac{1}{2}} + 7 x^{\frac{1}{2}} \cdot - 9 + 5 x^{\frac{1}{2}} + 5 \cdot - 9}{x - 81}$

Step 11

Simplify and combine like terms.

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Step 11.1

Simplify each term.

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Step 11.1.1

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

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Step 11.1.1.1

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

$- \frac{7 (x^{\frac{1}{2}} x^{\frac{1}{2}}) + 7 x^{\frac{1}{2}} \cdot - 9 + 5 x^{\frac{1}{2}} + 5 \cdot - 9}{x - 81}$

Step 11.1.1.2

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

$- \frac{7 x^{\frac{1}{2} + \frac{1}{2}} + 7 x^{\frac{1}{2}} \cdot - 9 + 5 x^{\frac{1}{2}} + 5 \cdot - 9}{x - 81}$

Step 11.1.1.3

Combine the numerators over the common denominator.

$- \frac{7 x^{\frac{1 + 1}{2}} + 7 x^{\frac{1}{2}} \cdot - 9 + 5 x^{\frac{1}{2}} + 5 \cdot - 9}{x - 81}$

Step 11.1.1.4

Add $1$ and $1$ .

$- \frac{7 x^{\frac{2}{2}} + 7 x^{\frac{1}{2}} \cdot - 9 + 5 x^{\frac{1}{2}} + 5 \cdot - 9}{x - 81}$

Step 11.1.1.5

Divide $2$ by $2$ .

$- \frac{7 x^{1} + 7 x^{\frac{1}{2}} \cdot - 9 + 5 x^{\frac{1}{2}} + 5 \cdot - 9}{x - 81}$

Step 11.1.2

Simplify $7 x^{1}$ .

$- \frac{7 x + 7 x^{\frac{1}{2}} \cdot - 9 + 5 x^{\frac{1}{2}} + 5 \cdot - 9}{x - 81}$

Step 11.1.3

Multiply $- 9$ by $7$ .

$- \frac{7 x - 63 x^{\frac{1}{2}} + 5 x^{\frac{1}{2}} + 5 \cdot - 9}{x - 81}$

Step 11.1.4

Multiply $5$ by $- 9$ .

$- \frac{7 x - 63 x^{\frac{1}{2}} + 5 x^{\frac{1}{2}} - 45}{x - 81}$

Step 11.2

Add $- 63 x^{\frac{1}{2}}$ and $5 x^{\frac{1}{2}}$ .

$- \frac{7 x - 58 x^{\frac{1}{2}} - 45}{x - 81}$

Step 12

Split the fraction $\frac{7 x - 58 x^{\frac{1}{2}} - 45}{x - 81}$ into two fractions.

$- (\frac{7 x - 58 x^{\frac{1}{2}}}{x - 81} + \frac{- 45}{x - 81})$

Step 13

Split the fraction $\frac{7 x - 58 x^{\frac{1}{2}}}{x - 81}$ into two fractions.

$- (\frac{7 x}{x - 81} + \frac{- 58 x^{\frac{1}{2}}}{x - 81} + \frac{- 45}{x - 81})$

Step 14

Move the negative in front of the fraction.

$- (\frac{7 x}{x - 81} - \frac{58 x^{\frac{1}{2}}}{x - 81} + \frac{- 45}{x - 81})$

Step 15

Move the negative in front of the fraction.

$- (\frac{7 x}{x - 81} - \frac{58 x^{\frac{1}{2}}}{x - 81} - \frac{45}{x - 81})$