Basic Math Examples

$\frac{5 \sqrt{c} + 2}{3 + \sqrt{c}}$

Step 1

Multiply $\frac{5 \sqrt{c} + 2}{3 + \sqrt{c}}$ by $\frac{3 - \sqrt{c}}{3 - \sqrt{c}}$ .

$\frac{5 \sqrt{c} + 2}{3 + \sqrt{c}} \cdot \frac{3 - \sqrt{c}}{3 - \sqrt{c}}$

Step 2

Combine fractions.

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Step 2.1

Multiply $\frac{5 \sqrt{c} + 2}{3 + \sqrt{c}}$ by $\frac{3 - \sqrt{c}}{3 - \sqrt{c}}$ .

$\frac{(5 \sqrt{c} + 2) (3 - \sqrt{c})}{(3 + \sqrt{c}) (3 - \sqrt{c})}$

Step 2.2

Expand the denominator using the FOIL method.

$\frac{(5 \sqrt{c} + 2) (3 - \sqrt{c})}{9 - 3 \sqrt{c} + \sqrt{c} \cdot 3 - {\sqrt{c}}^{2}}$

Step 2.3

Simplify.

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Step 2.3.1

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

$\frac{(5 \sqrt{c} + 2) (3 - \sqrt{c})}{9 - {(c^{\frac{1}{2}})}^{2}}$

Step 2.3.2

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

$\frac{(5 \sqrt{c} + 2) (3 - \sqrt{c})}{9 - c^{\frac{1}{2} \cdot 2}}$

Step 2.3.3

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

$\frac{(5 \sqrt{c} + 2) (3 - \sqrt{c})}{9 - c^{\frac{2}{2}}}$

Step 2.3.4

Cancel the common factor of $2$ .

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Step 2.3.4.1

Cancel the common factor.

$\frac{(5 \sqrt{c} + 2) (3 - \sqrt{c})}{9 - c^{\frac{2}{2}}}$

Step 2.3.4.2

Rewrite the expression.

$\frac{(5 \sqrt{c} + 2) (3 - \sqrt{c})}{9 - c^{1}}$

Step 2.3.5

Simplify.

$\frac{(5 \sqrt{c} + 2) (3 - \sqrt{c})}{9 - c}$

Step 3

Expand $(5 \sqrt{c} + 2) (3 - \sqrt{c})$ using the FOIL Method.

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Step 3.1

Apply the distributive property.

$\frac{5 \sqrt{c} (3 - \sqrt{c}) + 2 (3 - \sqrt{c})}{9 - c}$

Step 3.2

Apply the distributive property.

$\frac{5 \sqrt{c} \cdot 3 + 5 \sqrt{c} (- \sqrt{c}) + 2 (3 - \sqrt{c})}{9 - c}$

Step 3.3

Apply the distributive property.

$\frac{5 \sqrt{c} \cdot 3 + 5 \sqrt{c} (- \sqrt{c}) + 2 \cdot 3 + 2 (- \sqrt{c})}{9 - c}$

Step 4

Simplify and combine like terms.

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Step 4.1

Simplify each term.

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Step 4.1.1

Multiply $3$ by $5$ .

$\frac{15 \sqrt{c} + 5 \sqrt{c} (- \sqrt{c}) + 2 \cdot 3 + 2 (- \sqrt{c})}{9 - c}$

Step 4.1.2

Multiply $5 \sqrt{c} (- \sqrt{c})$ .

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Step 4.1.2.1

Multiply $- 1$ by $5$ .

$\frac{15 \sqrt{c} - 5 \sqrt{c} \sqrt{c} + 2 \cdot 3 + 2 (- \sqrt{c})}{9 - c}$

Step 4.1.2.2

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

$\frac{15 \sqrt{c} - 5 ({\sqrt{c}}^{1} \sqrt{c}) + 2 \cdot 3 + 2 (- \sqrt{c})}{9 - c}$

Step 4.1.2.3

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

$\frac{15 \sqrt{c} - 5 ({\sqrt{c}}^{1} {\sqrt{c}}^{1}) + 2 \cdot 3 + 2 (- \sqrt{c})}{9 - c}$

Step 4.1.2.4

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

$\frac{15 \sqrt{c} - 5 {\sqrt{c}}^{1 + 1} + 2 \cdot 3 + 2 (- \sqrt{c})}{9 - c}$

Step 4.1.2.5

Add $1$ and $1$ .

$\frac{15 \sqrt{c} - 5 {\sqrt{c}}^{2} + 2 \cdot 3 + 2 (- \sqrt{c})}{9 - c}$

Step 4.1.3

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

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Step 4.1.3.1

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

$\frac{15 \sqrt{c} - 5 {(c^{\frac{1}{2}})}^{2} + 2 \cdot 3 + 2 (- \sqrt{c})}{9 - c}$

Step 4.1.3.2

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

$\frac{15 \sqrt{c} - 5 c^{\frac{1}{2} \cdot 2} + 2 \cdot 3 + 2 (- \sqrt{c})}{9 - c}$

Step 4.1.3.3

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

$\frac{15 \sqrt{c} - 5 c^{\frac{2}{2}} + 2 \cdot 3 + 2 (- \sqrt{c})}{9 - c}$

Step 4.1.3.4

Cancel the common factor of $2$ .

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Step 4.1.3.4.1

Cancel the common factor.

$\frac{15 \sqrt{c} - 5 c^{\frac{2}{2}} + 2 \cdot 3 + 2 (- \sqrt{c})}{9 - c}$

Step 4.1.3.4.2

Rewrite the expression.

$\frac{15 \sqrt{c} - 5 c^{1} + 2 \cdot 3 + 2 (- \sqrt{c})}{9 - c}$

Step 4.1.3.5

Simplify.

$\frac{15 \sqrt{c} - 5 c + 2 \cdot 3 + 2 (- \sqrt{c})}{9 - c}$

Step 4.1.4

Multiply $2$ by $3$ .

$\frac{15 \sqrt{c} - 5 c + 6 + 2 (- \sqrt{c})}{9 - c}$

Step 4.1.5

Multiply $- 1$ by $2$ .

$\frac{15 \sqrt{c} - 5 c + 6 - 2 \sqrt{c}}{9 - c}$

Step 4.2

Subtract $2 \sqrt{c}$ from $15 \sqrt{c}$ .

$\frac{- 5 c + 6 + 13 \sqrt{c}}{9 - c}$

Step 5

Simplify the numerator.

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Step 5.1

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

$\frac{- 5 c + 6 + 13 c^{\frac{1}{2}}}{9 - c}$

Step 5.2

Reorder terms.

$\frac{- 5 c + 13 c^{\frac{1}{2}} + 6}{9 - c}$

Step 5.3

Rewrite $- 5 c + 13 c^{\frac{1}{2}} + 6$ in a factored form.

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Step 5.3.1

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

$\frac{- 5 {(c^{\frac{1}{2}})}^{2} + 13 c^{\frac{1}{2}} + 6}{9 - c}$

Step 5.3.2

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

$\frac{- 5 u^{2} + 13 u + 6}{9 - c}$

Step 5.3.3

Factor by grouping.

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Step 5.3.3.1

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 $a \cdot c = - 5 \cdot 6 = - 30$ and whose sum is $b = 13$ .

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Step 5.3.3.1.1

Factor $13$ out of $13 u$ .

$\frac{- 5 u^{2} + 13 (u) + 6}{9 - c}$

Step 5.3.3.1.2

Rewrite $13$ as $- 2$ plus $15$

$\frac{- 5 u^{2} + (- 2 + 15) u + 6}{9 - c}$

Step 5.3.3.1.3

Apply the distributive property.

$\frac{- 5 u^{2} - 2 u + 15 u + 6}{9 - c}$

Step 5.3.3.2

Factor out the greatest common factor from each group.

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Step 5.3.3.2.1

Group the first two terms and the last two terms.

$\frac{(- 5 u^{2} - 2 u) + 15 u + 6}{9 - c}$

Step 5.3.3.2.2

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

$\frac{u (- 5 u - 2) - 3 (- 5 u - 2)}{9 - c}$

Step 5.3.3.3

Factor the polynomial by factoring out the greatest common factor, $- 5 u - 2$ .

$\frac{(- 5 u - 2) (u - 3)}{9 - c}$

Step 5.3.4

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

$\frac{(- 5 c^{\frac{1}{2}} - 2) (c^{\frac{1}{2}} - 3)}{9 - c}$

Step 5.3.5

Factor $- 1$ out of $- 5 c^{\frac{1}{2}} - 2$ .

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Step 5.3.5.1

Factor $- 1$ out of $- 5 c^{\frac{1}{2}}$ .

$\frac{(- (5 c^{\frac{1}{2}}) - 2) (c^{\frac{1}{2}} - 3)}{9 - c}$

Step 5.3.5.2

Rewrite $- 2$ as $- 1 (2)$ .

$\frac{(- (5 c^{\frac{1}{2}}) - 1 (2)) (c^{\frac{1}{2}} - 3)}{9 - c}$

Step 5.3.5.3

Factor $- 1$ out of $- (5 c^{\frac{1}{2}}) - 1 (2)$ .

$\frac{- (5 c^{\frac{1}{2}} + 2) (c^{\frac{1}{2}} - 3)}{9 - c}$

Step 6

Move the negative in front of the fraction.

$- \frac{(5 c^{\frac{1}{2}} + 2) (c^{\frac{1}{2}} - 3)}{9 - c}$