Simplify ( square root of 7- square root of 5)/( square root of 12+ square root of 7)

$\frac{\sqrt{7} - \sqrt{5}}{\sqrt{12} + \sqrt{7}}$

Step 1

Simplify the denominator.

Rewrite $12$ as $2^{2} \cdot 3$ .

Factor $4$ out of $12$ .

$\frac{\sqrt{7} - \sqrt{5}}{\sqrt{4 (3)} + \sqrt{7}}$

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

$\frac{\sqrt{7} - \sqrt{5}}{\sqrt{2^{2} \cdot 3} + \sqrt{7}}$

Pull terms out from under the radical.

$\frac{\sqrt{7} - \sqrt{5}}{2 \sqrt{3} + \sqrt{7}}$

Step 2

Multiply $\frac{\sqrt{7} - \sqrt{5}}{2 \sqrt{3} + \sqrt{7}}$ by $\frac{2 \sqrt{3} - \sqrt{7}}{2 \sqrt{3} - \sqrt{7}}$ .

$\frac{\sqrt{7} - \sqrt{5}}{2 \sqrt{3} + \sqrt{7}} \cdot \frac{2 \sqrt{3} - \sqrt{7}}{2 \sqrt{3} - \sqrt{7}}$

Step 3

Combine fractions.

Multiply $\frac{\sqrt{7} - \sqrt{5}}{2 \sqrt{3} + \sqrt{7}}$ by $\frac{2 \sqrt{3} - \sqrt{7}}{2 \sqrt{3} - \sqrt{7}}$ .

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

Expand the denominator using the FOIL method.

$\frac{(\sqrt{7} - \sqrt{5}) (2 \sqrt{3} - \sqrt{7})}{4 {\sqrt{3}}^{2} - 2 \sqrt{21} + 2 \sqrt{21} - {\sqrt{7}}^{2}}$

Simplify.

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

Step 4

Expand $(\sqrt{7} - \sqrt{5}) (2 \sqrt{3} - \sqrt{7})$ using the FOIL Method.

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Step 5

Simplify each term.

Multiply $\sqrt{7} (2 \sqrt{3})$ .

Combine using the product rule for radicals.

$\frac{2 \sqrt{7 \cdot 3} + \sqrt{7} (- \sqrt{7}) - \sqrt{5} (2 \sqrt{3}) - \sqrt{5} (- \sqrt{7})}{5}$

Multiply $7$ by $3$ .

$\frac{2 \sqrt{21} + \sqrt{7} (- \sqrt{7}) - \sqrt{5} (2 \sqrt{3}) - \sqrt{5} (- \sqrt{7})}{5}$

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Evaluate the exponent.

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

Multiply $- 1$ by $7$ .

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

Multiply $- \sqrt{5} (2 \sqrt{3})$ .

Multiply $2$ by $- 1$ .

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

Combine using the product rule for radicals.

$\frac{2 \sqrt{21} - 7 - 2 \sqrt{3 \cdot 5} - \sqrt{5} (- \sqrt{7})}{5}$

Multiply $3$ by $5$ .

$\frac{2 \sqrt{21} - 7 - 2 \sqrt{15} - \sqrt{5} (- \sqrt{7})}{5}$

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

Multiply $- 1$ by $- 1$ .

$\frac{2 \sqrt{21} - 7 - 2 \sqrt{15} + 1 \sqrt{5} \sqrt{7}}{5}$

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

$\frac{2 \sqrt{21} - 7 - 2 \sqrt{15} + \sqrt{5} \sqrt{7}}{5}$

Combine using the product rule for radicals.

$\frac{2 \sqrt{21} - 7 - 2 \sqrt{15} + \sqrt{5 \cdot 7}}{5}$

Multiply $5$ by $7$ .

$\frac{2 \sqrt{21} - 7 - 2 \sqrt{15} + \sqrt{35}}{5}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\frac{2 \sqrt{21} - 7 - 2 \sqrt{15} + \sqrt{35}}{5}$

Decimal Form:

$0.06705289 \dots$

Simplify ( square root of 7- square root of 5)/( square root of 12+ square root of 7)

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