Simplify ( square root of 7- square root of 3)/( square root of 7+ square root of 3)

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

Step 1

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

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

Step 2

Combine fractions.

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

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

Expand the denominator using the FOIL method.

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

Simplify.

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

Step 3

Simplify the numerator.

Raise $\sqrt{7} - \sqrt{3}$ to the power of $1$ .

$\frac{{(\sqrt{7} - \sqrt{3})}^{1} (\sqrt{7} - \sqrt{3})}{4}$

Raise $\sqrt{7} - \sqrt{3}$ to the power of $1$ .

$\frac{{(\sqrt{7} - \sqrt{3})}^{1} {(\sqrt{7} - \sqrt{3})}^{1}}{4}$

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

$\frac{{(\sqrt{7} - \sqrt{3})}^{1 + 1}}{4}$

Add $1$ and $1$ .

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

Step 4

Rewrite ${(\sqrt{7} - \sqrt{3})}^{2}$ as $(\sqrt{7} - \sqrt{3}) (\sqrt{7} - \sqrt{3})$ .

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

Step 5

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Step 6

Simplify and combine like terms.

Simplify each term.

Combine using the product rule for radicals.

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

Multiply $7$ by $7$ .

$\frac{\sqrt{49} + \sqrt{7} (- \sqrt{3}) - \sqrt{3} \sqrt{7} - \sqrt{3} (- \sqrt{3})}{4}$

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

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

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

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

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

Combine using the product rule for radicals.

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

Multiply $7$ by $3$ .

$\frac{7 - \sqrt{21} - \sqrt{3} \sqrt{7} - \sqrt{3} (- \sqrt{3})}{4}$

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

Combine using the product rule for radicals.

$\frac{7 - \sqrt{21} - \sqrt{7 \cdot 3} - \sqrt{3} (- \sqrt{3})}{4}$

Multiply $7$ by $3$ .

$\frac{7 - \sqrt{21} - \sqrt{21} - \sqrt{3} (- \sqrt{3})}{4}$

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

Multiply $- 1$ by $- 1$ .

$\frac{7 - \sqrt{21} - \sqrt{21} + 1 \sqrt{3} \sqrt{3}}{4}$

Multiply $\sqrt{3}$ by $1$ .

$\frac{7 - \sqrt{21} - \sqrt{21} + \sqrt{3} \sqrt{3}}{4}$

Raise $\sqrt{3}$ to the power of $1$ .

$\frac{7 - \sqrt{21} - \sqrt{21} + {\sqrt{3}}^{1} \sqrt{3}}{4}$

Raise $\sqrt{3}$ to the power of $1$ .

$\frac{7 - \sqrt{21} - \sqrt{21} + {\sqrt{3}}^{1} {\sqrt{3}}^{1}}{4}$

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

$\frac{7 - \sqrt{21} - \sqrt{21} + {\sqrt{3}}^{1 + 1}}{4}$

Add $1$ and $1$ .

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

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$\frac{7 - \sqrt{21} - \sqrt{21} + {(3^{\frac{1}{2}})}^{2}}{4}$

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

$\frac{7 - \sqrt{21} - \sqrt{21} + 3^{\frac{1}{2} \cdot 2}}{4}$

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

$\frac{7 - \sqrt{21} - \sqrt{21} + 3^{1}}{4}$

Evaluate the exponent.

$\frac{7 - \sqrt{21} - \sqrt{21} + 3}{4}$

Add $7$ and $3$ .

$\frac{10 - \sqrt{21} - \sqrt{21}}{4}$

Subtract $\sqrt{21}$ from $- \sqrt{21}$ .

$\frac{10 - 2 \sqrt{21}}{4}$

Step 7

Cancel the common factor of $10 - 2 \sqrt{21}$ and $4$ .

Factor $2$ out of $10$ .

$\frac{2 (5) - 2 \sqrt{21}}{4}$

Factor $2$ out of $- 2 \sqrt{21}$ .

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

Factor $2$ out of $2 (5) + 2 (- \sqrt{21})$ .

$\frac{2 (5 - \sqrt{21})}{4}$

Cancel the common factors.

Factor $2$ out of $4$ .

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

Cancel the common factor.

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

Rewrite the expression.

$\frac{5 - \sqrt{21}}{2}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$\frac{5 - \sqrt{21}}{2}$

Decimal Form:

$0.20871215 \dots$

Simplify ( square root of 7- square root of 3)/( square root of 7+ square root of 3)

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