Algebra Examples

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

Step 1

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

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

Step 2

Combine fractions.

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Step 2.1

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

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

Step 2.2

Expand the denominator using the FOIL method.

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

Step 2.3

Simplify.

$\frac{(\sqrt{7} - \sqrt{5}) (\sqrt{7} - \sqrt{5})}{2}$

Step 3

Simplify the numerator.

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Step 3.1

Raise $\sqrt{7} - \sqrt{5}$ to the power of $1$ .

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

Step 3.2

Raise $\sqrt{7} - \sqrt{5}$ to the power of $1$ .

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

Step 3.3

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

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

Step 3.4

Add $1$ and $1$ .

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

Step 4

Rewrite ${(\sqrt{7} - \sqrt{5})}^{2}$ as $(\sqrt{7} - \sqrt{5}) (\sqrt{7} - \sqrt{5})$ .

$\frac{(\sqrt{7} - \sqrt{5}) (\sqrt{7} - \sqrt{5})}{2}$

Step 5

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

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Step 5.1

Apply the distributive property.

$\frac{\sqrt{7} (\sqrt{7} - \sqrt{5}) - \sqrt{5} (\sqrt{7} - \sqrt{5})}{2}$

Step 5.2

Apply the distributive property.

$\frac{\sqrt{7} \sqrt{7} + \sqrt{7} (- \sqrt{5}) - \sqrt{5} (\sqrt{7} - \sqrt{5})}{2}$

Step 5.3

Apply the distributive property.

$\frac{\sqrt{7} \sqrt{7} + \sqrt{7} (- \sqrt{5}) - \sqrt{5} \sqrt{7} - \sqrt{5} (- \sqrt{5})}{2}$

Step 6

Simplify and combine like terms.

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Step 6.1

Simplify each term.

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Step 6.1.1

Combine using the product rule for radicals.

$\frac{\sqrt{7 \cdot 7} + \sqrt{7} (- \sqrt{5}) - \sqrt{5} \sqrt{7} - \sqrt{5} (- \sqrt{5})}{2}$

Step 6.1.2

Multiply $7$ by $7$ .

$\frac{\sqrt{49} + \sqrt{7} (- \sqrt{5}) - \sqrt{5} \sqrt{7} - \sqrt{5} (- \sqrt{5})}{2}$

Step 6.1.3

Rewrite $49$ as $7^{2}$ .

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

Step 6.1.4

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

$\frac{7 + \sqrt{7} (- \sqrt{5}) - \sqrt{5} \sqrt{7} - \sqrt{5} (- \sqrt{5})}{2}$

Step 6.1.5

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

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Step 6.1.5.1

Combine using the product rule for radicals.

$\frac{7 - \sqrt{7 \cdot 5} - \sqrt{5} \sqrt{7} - \sqrt{5} (- \sqrt{5})}{2}$

Step 6.1.5.2

Multiply $7$ by $5$ .

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

Step 6.1.6

Multiply $- \sqrt{5} \sqrt{7}$ .

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Step 6.1.6.1

Combine using the product rule for radicals.

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

Step 6.1.6.2

Multiply $7$ by $5$ .

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

Step 6.1.7

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

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Step 6.1.7.1

Multiply $- 1$ by $- 1$ .

$\frac{7 - \sqrt{35} - \sqrt{35} + 1 \sqrt{5} \sqrt{5}}{2}$

Step 6.1.7.2

Multiply $\sqrt{5}$ by $1$ .

$\frac{7 - \sqrt{35} - \sqrt{35} + \sqrt{5} \sqrt{5}}{2}$

Step 6.1.7.3

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

$\frac{7 - \sqrt{35} - \sqrt{35} + {\sqrt{5}}^{1} \sqrt{5}}{2}$

Step 6.1.7.4

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

$\frac{7 - \sqrt{35} - \sqrt{35} + {\sqrt{5}}^{1} {\sqrt{5}}^{1}}{2}$

Step 6.1.7.5

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

$\frac{7 - \sqrt{35} - \sqrt{35} + {\sqrt{5}}^{1 + 1}}{2}$

Step 6.1.7.6

Add $1$ and $1$ .

$\frac{7 - \sqrt{35} - \sqrt{35} + {\sqrt{5}}^{2}}{2}$

Step 6.1.8

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

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Step 6.1.8.1

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

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

Step 6.1.8.2

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

$\frac{7 - \sqrt{35} - \sqrt{35} + 5^{\frac{1}{2} \cdot 2}}{2}$

Step 6.1.8.3

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

$\frac{7 - \sqrt{35} - \sqrt{35} + 5^{\frac{2}{2}}}{2}$

Step 6.1.8.4

Cancel the common factor of $2$ .

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Step 6.1.8.4.1

Cancel the common factor.

$\frac{7 - \sqrt{35} - \sqrt{35} + 5^{\frac{2}{2}}}{2}$

Step 6.1.8.4.2

Rewrite the expression.

$\frac{7 - \sqrt{35} - \sqrt{35} + 5^{1}}{2}$

Step 6.1.8.5

Evaluate the exponent.

$\frac{7 - \sqrt{35} - \sqrt{35} + 5}{2}$

Step 6.2

Add $7$ and $5$ .

$\frac{12 - \sqrt{35} - \sqrt{35}}{2}$

Step 6.3

Subtract $\sqrt{35}$ from $- \sqrt{35}$ .

$\frac{12 - 2 \sqrt{35}}{2}$

Step 7

Cancel the common factor of $12 - 2 \sqrt{35}$ and $2$ .

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Step 7.1

Factor $2$ out of $12$ .

$\frac{2 \cdot 6 - 2 \sqrt{35}}{2}$

Step 7.2

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

$\frac{2 \cdot 6 + 2 (- \sqrt{35})}{2}$

Step 7.3

Factor $2$ out of $2 (6) + 2 (- \sqrt{35})$ .

$\frac{2 (6 - \sqrt{35})}{2}$

Step 7.4

Cancel the common factors.

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Step 7.4.1

Factor $2$ out of $2$ .

$\frac{2 (6 - \sqrt{35})}{2 (1)}$

Step 7.4.2

Cancel the common factor.

$\frac{2 (6 - \sqrt{35})}{2 \cdot 1}$

Step 7.4.3

Rewrite the expression.

$\frac{6 - \sqrt{35}}{1}$

Step 7.4.4

Divide $6 - \sqrt{35}$ by $1$ .

$6 - \sqrt{35}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$6 - \sqrt{35}$

Decimal Form:

$0.08392021 \dots$