Basic Math Examples

$\frac{\sqrt{60} - \sqrt{20}}{\sqrt{5} - \sqrt{15}}$

Step 1

Simplify the numerator.

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Step 1.1

Rewrite $60$ as $2^{2} \cdot 15$ .

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Step 1.1.1

Factor $4$ out of $60$ .

$\frac{\sqrt{4 (15)} - \sqrt{20}}{\sqrt{5} - \sqrt{15}}$

Step 1.1.2

Rewrite $4$ as $2^{2}$ .

$\frac{\sqrt{2^{2} \cdot 15} - \sqrt{20}}{\sqrt{5} - \sqrt{15}}$

Step 1.2

Pull terms out from under the radical.

$\frac{2 \sqrt{15} - \sqrt{20}}{\sqrt{5} - \sqrt{15}}$

Step 1.3

Rewrite $20$ as $2^{2} \cdot 5$ .

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Step 1.3.1

Factor $4$ out of $20$ .

$\frac{2 \sqrt{15} - \sqrt{4 (5)}}{\sqrt{5} - \sqrt{15}}$

Step 1.3.2

Rewrite $4$ as $2^{2}$ .

$\frac{2 \sqrt{15} - \sqrt{2^{2} \cdot 5}}{\sqrt{5} - \sqrt{15}}$

Step 1.4

Pull terms out from under the radical.

$\frac{2 \sqrt{15} - (2 \sqrt{5})}{\sqrt{5} - \sqrt{15}}$

Step 1.5

Multiply $2$ by $- 1$ .

$\frac{2 \sqrt{15} - 2 \sqrt{5}}{\sqrt{5} - \sqrt{15}}$

Step 2

Multiply $\frac{2 \sqrt{15} - 2 \sqrt{5}}{\sqrt{5} - \sqrt{15}}$ by $\frac{\sqrt{5} + \sqrt{15}}{\sqrt{5} + \sqrt{15}}$ .

$\frac{2 \sqrt{15} - 2 \sqrt{5}}{\sqrt{5} - \sqrt{15}} \cdot \frac{\sqrt{5} + \sqrt{15}}{\sqrt{5} + \sqrt{15}}$

Step 3

Simplify terms.

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Step 3.1

Multiply $\frac{2 \sqrt{15} - 2 \sqrt{5}}{\sqrt{5} - \sqrt{15}}$ by $\frac{\sqrt{5} + \sqrt{15}}{\sqrt{5} + \sqrt{15}}$ .

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

Step 3.2

Expand the denominator using the FOIL method.

$\frac{(2 \sqrt{15} - 2 \sqrt{5}) (\sqrt{5} + \sqrt{15})}{{\sqrt{5}}^{2} + \sqrt{75} - \sqrt{75} - {\sqrt{15}}^{2}}$

Step 3.3

Simplify.

$\frac{(2 \sqrt{15} - 2 \sqrt{5}) (\sqrt{5} + \sqrt{15})}{- 10}$

Step 3.4

Cancel the common factor of $2 \sqrt{15} - 2 \sqrt{5}$ and $- 10$ .

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Step 3.4.1

Factor $2$ out of $(2 \sqrt{15} - 2 \sqrt{5}) (\sqrt{5} + \sqrt{15})$ .

$\frac{2 ((\sqrt{15} - \sqrt{5}) (\sqrt{5} + \sqrt{15}))}{- 10}$

Step 3.4.2

Cancel the common factors.

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Step 3.4.2.1

Factor $2$ out of $- 10$ .

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

Step 3.4.2.2

Cancel the common factor.

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

Step 3.4.2.3

Rewrite the expression.

$\frac{(\sqrt{15} - \sqrt{5}) (\sqrt{5} + \sqrt{15})}{- 5}$

Step 4

Simplify the numerator.

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Step 4.1

Expand $(\sqrt{15} - \sqrt{5}) (\sqrt{5} + \sqrt{15})$ using the FOIL Method.

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Step 4.1.1

Apply the distributive property.

$\frac{\sqrt{15} (\sqrt{5} + \sqrt{15}) - \sqrt{5} (\sqrt{5} + \sqrt{15})}{- 5}$

Step 4.1.2

Apply the distributive property.

$\frac{\sqrt{15} \sqrt{5} + \sqrt{15} \sqrt{15} - \sqrt{5} (\sqrt{5} + \sqrt{15})}{- 5}$

Step 4.1.3

Apply the distributive property.

$\frac{\sqrt{15} \sqrt{5} + \sqrt{15} \sqrt{15} - \sqrt{5} \sqrt{5} - \sqrt{5} \sqrt{15}}{- 5}$

Step 4.2

Simplify and combine like terms.

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Step 4.2.1

Simplify each term.

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Step 4.2.1.1

Combine using the product rule for radicals.

$\frac{\sqrt{15 \cdot 5} + \sqrt{15} \sqrt{15} - \sqrt{5} \sqrt{5} - \sqrt{5} \sqrt{15}}{- 5}$

Step 4.2.1.2

Multiply $15$ by $5$ .

$\frac{\sqrt{75} + \sqrt{15} \sqrt{15} - \sqrt{5} \sqrt{5} - \sqrt{5} \sqrt{15}}{- 5}$

Step 4.2.1.3

Rewrite $75$ as $5^{2} \cdot 3$ .

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Step 4.2.1.3.1

Factor $25$ out of $75$ .

$\frac{\sqrt{25 (3)} + \sqrt{15} \sqrt{15} - \sqrt{5} \sqrt{5} - \sqrt{5} \sqrt{15}}{- 5}$

Step 4.2.1.3.2

Rewrite $25$ as $5^{2}$ .

$\frac{\sqrt{5^{2} \cdot 3} + \sqrt{15} \sqrt{15} - \sqrt{5} \sqrt{5} - \sqrt{5} \sqrt{15}}{- 5}$

Step 4.2.1.4

Pull terms out from under the radical.

$\frac{5 \sqrt{3} + \sqrt{15} \sqrt{15} - \sqrt{5} \sqrt{5} - \sqrt{5} \sqrt{15}}{- 5}$

Step 4.2.1.5

Combine using the product rule for radicals.

$\frac{5 \sqrt{3} + \sqrt{15 \cdot 15} - \sqrt{5} \sqrt{5} - \sqrt{5} \sqrt{15}}{- 5}$

Step 4.2.1.6

Multiply $15$ by $15$ .

$\frac{5 \sqrt{3} + \sqrt{225} - \sqrt{5} \sqrt{5} - \sqrt{5} \sqrt{15}}{- 5}$

Step 4.2.1.7

Rewrite $225$ as $15^{2}$ .

$\frac{5 \sqrt{3} + \sqrt{15^{2}} - \sqrt{5} \sqrt{5} - \sqrt{5} \sqrt{15}}{- 5}$

Step 4.2.1.8

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

$\frac{5 \sqrt{3} + 15 - \sqrt{5} \sqrt{5} - \sqrt{5} \sqrt{15}}{- 5}$

Step 4.2.1.9

Multiply $- \sqrt{5} \sqrt{5}$ .

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Step 4.2.1.9.1

Raise $\sqrt{5}$ to the power of $1$ .

$\frac{5 \sqrt{3} + 15 - ({\sqrt{5}}^{1} \sqrt{5}) - \sqrt{5} \sqrt{15}}{- 5}$

Step 4.2.1.9.2

Raise $\sqrt{5}$ to the power of $1$ .

$\frac{5 \sqrt{3} + 15 - ({\sqrt{5}}^{1} {\sqrt{5}}^{1}) - \sqrt{5} \sqrt{15}}{- 5}$

Step 4.2.1.9.3

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

$\frac{5 \sqrt{3} + 15 - {\sqrt{5}}^{1 + 1} - \sqrt{5} \sqrt{15}}{- 5}$

Step 4.2.1.9.4

Add $1$ and $1$ .

$\frac{5 \sqrt{3} + 15 - {\sqrt{5}}^{2} - \sqrt{5} \sqrt{15}}{- 5}$

Step 4.2.1.10

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

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Step 4.2.1.10.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

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

Step 4.2.1.10.2

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

$\frac{5 \sqrt{3} + 15 - 5^{\frac{1}{2} \cdot 2} - \sqrt{5} \sqrt{15}}{- 5}$

Step 4.2.1.10.3

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

$\frac{5 \sqrt{3} + 15 - 5^{\frac{2}{2}} - \sqrt{5} \sqrt{15}}{- 5}$

Step 4.2.1.10.4

Cancel the common factor of $2$ .

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Step 4.2.1.10.4.1

Cancel the common factor.

$\frac{5 \sqrt{3} + 15 - 5^{\frac{2}{2}} - \sqrt{5} \sqrt{15}}{- 5}$

Step 4.2.1.10.4.2

Rewrite the expression.

$\frac{5 \sqrt{3} + 15 - 5^{1} - \sqrt{5} \sqrt{15}}{- 5}$

Step 4.2.1.10.5

Evaluate the exponent.

$\frac{5 \sqrt{3} + 15 - 1 \cdot 5 - \sqrt{5} \sqrt{15}}{- 5}$

Step 4.2.1.11

Multiply $- 1$ by $5$ .

$\frac{5 \sqrt{3} + 15 - 5 - \sqrt{5} \sqrt{15}}{- 5}$

Step 4.2.1.12

Multiply $- \sqrt{5} \sqrt{15}$ .

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Step 4.2.1.12.1

Combine using the product rule for radicals.

$\frac{5 \sqrt{3} + 15 - 5 - \sqrt{15 \cdot 5}}{- 5}$

Step 4.2.1.12.2

Multiply $15$ by $5$ .

$\frac{5 \sqrt{3} + 15 - 5 - \sqrt{75}}{- 5}$

Step 4.2.1.13

Rewrite $75$ as $5^{2} \cdot 3$ .

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Step 4.2.1.13.1

Factor $25$ out of $75$ .

$\frac{5 \sqrt{3} + 15 - 5 - \sqrt{25 (3)}}{- 5}$

Step 4.2.1.13.2

Rewrite $25$ as $5^{2}$ .

$\frac{5 \sqrt{3} + 15 - 5 - \sqrt{5^{2} \cdot 3}}{- 5}$

Step 4.2.1.14

Pull terms out from under the radical.

$\frac{5 \sqrt{3} + 15 - 5 - (5 \sqrt{3})}{- 5}$

Step 4.2.1.15

Multiply $5$ by $- 1$ .

$\frac{5 \sqrt{3} + 15 - 5 - 5 \sqrt{3}}{- 5}$

Step 4.2.2

Subtract $5 \sqrt{3}$ from $5 \sqrt{3}$ .

$\frac{0 + 15 - 5}{- 5}$

Step 4.2.3

Add $0$ and $15$ .

$\frac{15 - 5}{- 5}$

Step 4.2.4

Subtract $5$ from $15$ .

$\frac{10}{- 5}$

Step 5

Divide $10$ by $- 5$ .

$- 2$