Simplify ( square root of 10- square root of 3)/( square root of 10+ square root of 3)

$\frac{\sqrt{10} - \sqrt{3}}{\sqrt{10} + \sqrt{3}}$

Step 1

Multiply $\frac{\sqrt{10} - \sqrt{3}}{\sqrt{10} + \sqrt{3}}$ by $\frac{\sqrt{10} - \sqrt{3}}{\sqrt{10} - \sqrt{3}}$ .

$\frac{\sqrt{10} - \sqrt{3}}{\sqrt{10} + \sqrt{3}} \cdot \frac{\sqrt{10} - \sqrt{3}}{\sqrt{10} - \sqrt{3}}$

Step 2

Combine fractions.

Multiply $\frac{\sqrt{10} - \sqrt{3}}{\sqrt{10} + \sqrt{3}}$ by $\frac{\sqrt{10} - \sqrt{3}}{\sqrt{10} - \sqrt{3}}$ .

$\frac{(\sqrt{10} - \sqrt{3}) (\sqrt{10} - \sqrt{3})}{(\sqrt{10} + \sqrt{3}) (\sqrt{10} - \sqrt{3})}$

Expand the denominator using the FOIL method.

$\frac{(\sqrt{10} - \sqrt{3}) (\sqrt{10} - \sqrt{3})}{{\sqrt{10}}^{2} - \sqrt{30} + \sqrt{30} - {\sqrt{3}}^{2}}$

Simplify.

$\frac{(\sqrt{10} - \sqrt{3}) (\sqrt{10} - \sqrt{3})}{7}$

Step 3

Simplify the numerator.

Raise $\sqrt{10} - \sqrt{3}$ to the power of $1$ .

$\frac{{(\sqrt{10} - \sqrt{3})}^{1} (\sqrt{10} - \sqrt{3})}{7}$

Raise $\sqrt{10} - \sqrt{3}$ to the power of $1$ .

$\frac{{(\sqrt{10} - \sqrt{3})}^{1} {(\sqrt{10} - \sqrt{3})}^{1}}{7}$

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

$\frac{{(\sqrt{10} - \sqrt{3})}^{1 + 1}}{7}$

Add $1$ and $1$ .

$\frac{{(\sqrt{10} - \sqrt{3})}^{2}}{7}$

Step 4

Rewrite ${(\sqrt{10} - \sqrt{3})}^{2}$ as $(\sqrt{10} - \sqrt{3}) (\sqrt{10} - \sqrt{3})$ .

$\frac{(\sqrt{10} - \sqrt{3}) (\sqrt{10} - \sqrt{3})}{7}$

Step 5

Expand $(\sqrt{10} - \sqrt{3}) (\sqrt{10} - \sqrt{3})$ using the FOIL Method.

Apply the distributive property.

$\frac{\sqrt{10} (\sqrt{10} - \sqrt{3}) - \sqrt{3} (\sqrt{10} - \sqrt{3})}{7}$

Apply the distributive property.

$\frac{\sqrt{10} \sqrt{10} + \sqrt{10} (- \sqrt{3}) - \sqrt{3} (\sqrt{10} - \sqrt{3})}{7}$

Apply the distributive property.

$\frac{\sqrt{10} \sqrt{10} + \sqrt{10} (- \sqrt{3}) - \sqrt{3} \sqrt{10} - \sqrt{3} (- \sqrt{3})}{7}$

Step 6

Simplify and combine like terms.

Simplify each term.

Combine using the product rule for radicals.

$\frac{\sqrt{10 \cdot 10} + \sqrt{10} (- \sqrt{3}) - \sqrt{3} \sqrt{10} - \sqrt{3} (- \sqrt{3})}{7}$

Multiply $10$ by $10$ .

$\frac{\sqrt{100} + \sqrt{10} (- \sqrt{3}) - \sqrt{3} \sqrt{10} - \sqrt{3} (- \sqrt{3})}{7}$

Rewrite $100$ as $10^{2}$ .

$\frac{\sqrt{10^{2}} + \sqrt{10} (- \sqrt{3}) - \sqrt{3} \sqrt{10} - \sqrt{3} (- \sqrt{3})}{7}$

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

$\frac{10 + \sqrt{10} (- \sqrt{3}) - \sqrt{3} \sqrt{10} - \sqrt{3} (- \sqrt{3})}{7}$

Multiply $\sqrt{10} (- \sqrt{3})$ .

Combine using the product rule for radicals.

$\frac{10 - \sqrt{10 \cdot 3} - \sqrt{3} \sqrt{10} - \sqrt{3} (- \sqrt{3})}{7}$

Multiply $10$ by $3$ .

$\frac{10 - \sqrt{30} - \sqrt{3} \sqrt{10} - \sqrt{3} (- \sqrt{3})}{7}$

Multiply $- \sqrt{3} \sqrt{10}$ .

Combine using the product rule for radicals.

$\frac{10 - \sqrt{30} - \sqrt{10 \cdot 3} - \sqrt{3} (- \sqrt{3})}{7}$

Multiply $10$ by $3$ .

$\frac{10 - \sqrt{30} - \sqrt{30} - \sqrt{3} (- \sqrt{3})}{7}$

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

Multiply $- 1$ by $- 1$ .

$\frac{10 - \sqrt{30} - \sqrt{30} + 1 \sqrt{3} \sqrt{3}}{7}$

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

$\frac{10 - \sqrt{30} - \sqrt{30} + \sqrt{3} \sqrt{3}}{7}$

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

$\frac{10 - \sqrt{30} - \sqrt{30} + {\sqrt{3}}^{1} \sqrt{3}}{7}$

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

$\frac{10 - \sqrt{30} - \sqrt{30} + {\sqrt{3}}^{1} {\sqrt{3}}^{1}}{7}$

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

$\frac{10 - \sqrt{30} - \sqrt{30} + {\sqrt{3}}^{1 + 1}}{7}$

Add $1$ and $1$ .

$\frac{10 - \sqrt{30} - \sqrt{30} + {\sqrt{3}}^{2}}{7}$

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

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

$\frac{10 - \sqrt{30} - \sqrt{30} + {(3^{\frac{1}{2}})}^{2}}{7}$

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

$\frac{10 - \sqrt{30} - \sqrt{30} + 3^{\frac{1}{2} \cdot 2}}{7}$

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

$\frac{10 - \sqrt{30} - \sqrt{30} + 3^{\frac{2}{2}}}{7}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{10 - \sqrt{30} - \sqrt{30} + 3^{\frac{2}{2}}}{7}$

Rewrite the expression.

$\frac{10 - \sqrt{30} - \sqrt{30} + 3^{1}}{7}$

Evaluate the exponent.

$\frac{10 - \sqrt{30} - \sqrt{30} + 3}{7}$

Add $10$ and $3$ .

$\frac{13 - \sqrt{30} - \sqrt{30}}{7}$

Subtract $\sqrt{30}$ from $- \sqrt{30}$ .

$\frac{13 - 2 \sqrt{30}}{7}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$\frac{13 - 2 \sqrt{30}}{7}$

Decimal Form:

$0.29222126 \dots$

Simplify ( square root of 10- square root of 3)/( square root of 10+ square root of 3)

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