Precalculus Examples

$\frac{\sqrt{7} - \sqrt{5}}{\sqrt{5} + 3 \sqrt{7}}$

Step 1

Multiply $\frac{\sqrt{7} - \sqrt{5}}{\sqrt{5} + 3 \sqrt{7}}$ by $\frac{\sqrt{5} - 3 \sqrt{7}}{\sqrt{5} - 3 \sqrt{7}}$ .

$\frac{\sqrt{7} - \sqrt{5}}{\sqrt{5} + 3 \sqrt{7}} \cdot \frac{\sqrt{5} - 3 \sqrt{7}}{\sqrt{5} - 3 \sqrt{7}}$

Step 2

Combine fractions.

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Step 2.1

Multiply $\frac{\sqrt{7} - \sqrt{5}}{\sqrt{5} + 3 \sqrt{7}}$ by $\frac{\sqrt{5} - 3 \sqrt{7}}{\sqrt{5} - 3 \sqrt{7}}$ .

$\frac{(\sqrt{7} - \sqrt{5}) (\sqrt{5} - 3 \sqrt{7})}{(\sqrt{5} + 3 \sqrt{7}) (\sqrt{5} - 3 \sqrt{7})}$

Step 2.2

Expand the denominator using the FOIL method.

$\frac{(\sqrt{7} - \sqrt{5}) (\sqrt{5} - 3 \sqrt{7})}{{\sqrt{5}}^{2} - 3 \sqrt{35} + 3 \sqrt{35} - 9 {\sqrt{7}}^{2}}$

Step 2.3

Simplify.

$\frac{(\sqrt{7} - \sqrt{5}) (\sqrt{5} - 3 \sqrt{7})}{- 58}$

Step 3

Expand $(\sqrt{7} - \sqrt{5}) (\sqrt{5} - 3 \sqrt{7})$ using the FOIL Method.

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Step 3.1

Apply the distributive property.

$\frac{\sqrt{7} (\sqrt{5} - 3 \sqrt{7}) - \sqrt{5} (\sqrt{5} - 3 \sqrt{7})}{- 58}$

Step 3.2

Apply the distributive property.

$\frac{\sqrt{7} \sqrt{5} + \sqrt{7} (- 3 \sqrt{7}) - \sqrt{5} (\sqrt{5} - 3 \sqrt{7})}{- 58}$

Step 3.3

Apply the distributive property.

$\frac{\sqrt{7} \sqrt{5} + \sqrt{7} (- 3 \sqrt{7}) - \sqrt{5} \sqrt{5} - \sqrt{5} (- 3 \sqrt{7})}{- 58}$

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

Combine using the product rule for radicals.

$\frac{\sqrt{7 \cdot 5} + \sqrt{7} (- 3 \sqrt{7}) - \sqrt{5} \sqrt{5} - \sqrt{5} (- 3 \sqrt{7})}{- 58}$

Step 4.1.2

Multiply $7$ by $5$ .

$\frac{\sqrt{35} + \sqrt{7} (- 3 \sqrt{7}) - \sqrt{5} \sqrt{5} - \sqrt{5} (- 3 \sqrt{7})}{- 58}$

Step 4.1.3

Multiply $\sqrt{7} (- 3 \sqrt{7})$ .

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Step 4.1.3.1

Raise $\sqrt{7}$ to the power of $1$ .

$\frac{\sqrt{35} - 3 ({\sqrt{7}}^{1} \sqrt{7}) - \sqrt{5} \sqrt{5} - \sqrt{5} (- 3 \sqrt{7})}{- 58}$

Step 4.1.3.2

Raise $\sqrt{7}$ to the power of $1$ .

$\frac{\sqrt{35} - 3 ({\sqrt{7}}^{1} {\sqrt{7}}^{1}) - \sqrt{5} \sqrt{5} - \sqrt{5} (- 3 \sqrt{7})}{- 58}$

Step 4.1.3.3

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

$\frac{\sqrt{35} - 3 {\sqrt{7}}^{1 + 1} - \sqrt{5} \sqrt{5} - \sqrt{5} (- 3 \sqrt{7})}{- 58}$

Step 4.1.3.4

Add $1$ and $1$ .

$\frac{\sqrt{35} - 3 {\sqrt{7}}^{2} - \sqrt{5} \sqrt{5} - \sqrt{5} (- 3 \sqrt{7})}{- 58}$

Step 4.1.4

Rewrite ${\sqrt{7}}^{2}$ as $7$ .

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Step 4.1.4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{7}$ as $7^{\frac{1}{2}}$ .

$\frac{\sqrt{35} - 3 {(7^{\frac{1}{2}})}^{2} - \sqrt{5} \sqrt{5} - \sqrt{5} (- 3 \sqrt{7})}{- 58}$

Step 4.1.4.2

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

$\frac{\sqrt{35} - 3 \cdot 7^{\frac{1}{2} \cdot 2} - \sqrt{5} \sqrt{5} - \sqrt{5} (- 3 \sqrt{7})}{- 58}$

Step 4.1.4.3

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

$\frac{\sqrt{35} - 3 \cdot 7^{\frac{2}{2}} - \sqrt{5} \sqrt{5} - \sqrt{5} (- 3 \sqrt{7})}{- 58}$

Step 4.1.4.4

Cancel the common factor of $2$ .

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Step 4.1.4.4.1

Cancel the common factor.

$\frac{\sqrt{35} - 3 \cdot 7^{\frac{2}{2}} - \sqrt{5} \sqrt{5} - \sqrt{5} (- 3 \sqrt{7})}{- 58}$

Step 4.1.4.4.2

Rewrite the expression.

$\frac{\sqrt{35} - 3 \cdot 7^{1} - \sqrt{5} \sqrt{5} - \sqrt{5} (- 3 \sqrt{7})}{- 58}$

Step 4.1.4.5

Evaluate the exponent.

$\frac{\sqrt{35} - 3 \cdot 7 - \sqrt{5} \sqrt{5} - \sqrt{5} (- 3 \sqrt{7})}{- 58}$

Step 4.1.5

Multiply $- 3$ by $7$ .

$\frac{\sqrt{35} - 21 - \sqrt{5} \sqrt{5} - \sqrt{5} (- 3 \sqrt{7})}{- 58}$

Step 4.1.6

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

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Step 4.1.6.1

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

$\frac{\sqrt{35} - 21 - ({\sqrt{5}}^{1} \sqrt{5}) - \sqrt{5} (- 3 \sqrt{7})}{- 58}$

Step 4.1.6.2

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

$\frac{\sqrt{35} - 21 - ({\sqrt{5}}^{1} {\sqrt{5}}^{1}) - \sqrt{5} (- 3 \sqrt{7})}{- 58}$

Step 4.1.6.3

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

$\frac{\sqrt{35} - 21 - {\sqrt{5}}^{1 + 1} - \sqrt{5} (- 3 \sqrt{7})}{- 58}$

Step 4.1.6.4

Add $1$ and $1$ .

$\frac{\sqrt{35} - 21 - {\sqrt{5}}^{2} - \sqrt{5} (- 3 \sqrt{7})}{- 58}$

Step 4.1.7

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

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Step 4.1.7.1

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

$\frac{\sqrt{35} - 21 - {(5^{\frac{1}{2}})}^{2} - \sqrt{5} (- 3 \sqrt{7})}{- 58}$

Step 4.1.7.2

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

$\frac{\sqrt{35} - 21 - 5^{\frac{1}{2} \cdot 2} - \sqrt{5} (- 3 \sqrt{7})}{- 58}$

Step 4.1.7.3

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

$\frac{\sqrt{35} - 21 - 5^{\frac{2}{2}} - \sqrt{5} (- 3 \sqrt{7})}{- 58}$

Step 4.1.7.4

Cancel the common factor of $2$ .

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Step 4.1.7.4.1

Cancel the common factor.

$\frac{\sqrt{35} - 21 - 5^{\frac{2}{2}} - \sqrt{5} (- 3 \sqrt{7})}{- 58}$

Step 4.1.7.4.2

Rewrite the expression.

$\frac{\sqrt{35} - 21 - 5^{1} - \sqrt{5} (- 3 \sqrt{7})}{- 58}$

Step 4.1.7.5

Evaluate the exponent.

$\frac{\sqrt{35} - 21 - 1 \cdot 5 - \sqrt{5} (- 3 \sqrt{7})}{- 58}$

Step 4.1.8

Multiply $- 1$ by $5$ .

$\frac{\sqrt{35} - 21 - 5 - \sqrt{5} (- 3 \sqrt{7})}{- 58}$

Step 4.1.9

Multiply $- \sqrt{5} (- 3 \sqrt{7})$ .

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Step 4.1.9.1

Multiply $- 3$ by $- 1$ .

$\frac{\sqrt{35} - 21 - 5 + 3 \sqrt{5} \sqrt{7}}{- 58}$

Step 4.1.9.2

Combine using the product rule for radicals.

$\frac{\sqrt{35} - 21 - 5 + 3 \sqrt{7 \cdot 5}}{- 58}$

Step 4.1.9.3

Multiply $7$ by $5$ .

$\frac{\sqrt{35} - 21 - 5 + 3 \sqrt{35}}{- 58}$

Step 4.2

Add $\sqrt{35}$ and $3 \sqrt{35}$ .

$\frac{- 21 - 5 + 4 \sqrt{35}}{- 58}$

Step 4.3

Subtract $5$ from $- 21$ .

$\frac{- 26 + 4 \sqrt{35}}{- 58}$

Step 5

Cancel the common factor of $- 26 + 4 \sqrt{35}$ and $- 58$ .

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Step 5.1

Factor $2$ out of $- 26$ .

$\frac{2 (- 13) + 4 \sqrt{35}}{- 58}$

Step 5.2

Factor $2$ out of $4 \sqrt{35}$ .

$\frac{2 (- 13) + 2 (2 \sqrt{35})}{- 58}$

Step 5.3

Factor $2$ out of $2 (- 13) + 2 (2 \sqrt{35})$ .

$\frac{2 (- 13 + 2 \sqrt{35})}{- 58}$

Step 5.4

Cancel the common factors.

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Step 5.4.1

Factor $2$ out of $- 58$ .

$\frac{2 (- 13 + 2 \sqrt{35})}{2 \cdot - 29}$

Step 5.4.2

Cancel the common factor.

$\frac{2 (- 13 + 2 \sqrt{35})}{2 \cdot - 29}$

Step 5.4.3

Rewrite the expression.

$\frac{- 13 + 2 \sqrt{35}}{- 29}$

Step 6

Move the negative in front of the fraction.

$- \frac{- 13 + 2 \sqrt{35}}{29}$

Step 7

Rewrite $- 13$ as $- 1 (13)$ .

$- \frac{- 1 (13) + 2 \sqrt{35}}{29}$

Step 8

Factor $- 1$ out of $2 \sqrt{35}$ .

$- \frac{- 1 (13) - (- 2 \sqrt{35})}{29}$

Step 9

Factor $- 1$ out of $- 1 (13) - (- 2 \sqrt{35})$ .

$- \frac{- 1 (13 - 2 \sqrt{35})}{29}$

Step 10

Simplify the expression.

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Step 10.1

Move the negative in front of the fraction.

$- - \frac{13 - 2 \sqrt{35}}{29}$

Step 10.2

Multiply $- 1$ by $- 1$ .

$1 \frac{13 - 2 \sqrt{35}}{29}$

Step 10.3

Multiply $\frac{13 - 2 \sqrt{35}}{29}$ by $1$ .

$\frac{13 - 2 \sqrt{35}}{29}$

Step 11

The result can be shown in multiple forms.

Exact Form:

$\frac{13 - 2 \sqrt{35}}{29}$

Decimal Form:

$0.04027035 \dots$