Evaluate ( square root of 20- square root of 3)( square root of 20+ square root of 3)

$(\sqrt{20} - \sqrt{3}) (\sqrt{20} + \sqrt{3})$

Step 1

Simplify terms.

Simplify each term.

Rewrite $20$ as $2^{2} \cdot 5$ .

Factor $4$ out of $20$ .

$(\sqrt{4 (5)} - \sqrt{3}) (\sqrt{20} + \sqrt{3})$

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

$(\sqrt{2^{2} \cdot 5} - \sqrt{3}) (\sqrt{20} + \sqrt{3})$

Pull terms out from under the radical.

$(2 \sqrt{5} - \sqrt{3}) (\sqrt{20} + \sqrt{3})$

Simplify each term.

Rewrite $20$ as $2^{2} \cdot 5$ .

Factor $4$ out of $20$ .

$(2 \sqrt{5} - \sqrt{3}) (\sqrt{4 (5)} + \sqrt{3})$

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

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

Pull terms out from under the radical.

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

Step 2

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Step 3

Simplify and combine like terms.

Simplify each term.

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

Multiply $2$ by $2$ .

$4 \sqrt{5} \sqrt{5} + 2 \sqrt{5} \sqrt{3} - \sqrt{3} (2 \sqrt{5}) - \sqrt{3} \sqrt{3}$

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

$4 ({\sqrt{5}}^{1} \sqrt{5}) + 2 \sqrt{5} \sqrt{3} - \sqrt{3} (2 \sqrt{5}) - \sqrt{3} \sqrt{3}$

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

$4 ({\sqrt{5}}^{1} {\sqrt{5}}^{1}) + 2 \sqrt{5} \sqrt{3} - \sqrt{3} (2 \sqrt{5}) - \sqrt{3} \sqrt{3}$

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

$4 {\sqrt{5}}^{1 + 1} + 2 \sqrt{5} \sqrt{3} - \sqrt{3} (2 \sqrt{5}) - \sqrt{3} \sqrt{3}$

Add $1$ and $1$ .

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

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

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

$4 {(5^{\frac{1}{2}})}^{2} + 2 \sqrt{5} \sqrt{3} - \sqrt{3} (2 \sqrt{5}) - \sqrt{3} \sqrt{3}$

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

$4 \cdot 5^{\frac{1}{2} \cdot 2} + 2 \sqrt{5} \sqrt{3} - \sqrt{3} (2 \sqrt{5}) - \sqrt{3} \sqrt{3}$

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

$4 \cdot 5^{1} + 2 \sqrt{5} \sqrt{3} - \sqrt{3} (2 \sqrt{5}) - \sqrt{3} \sqrt{3}$

Evaluate the exponent.

$4 \cdot 5 + 2 \sqrt{5} \sqrt{3} - \sqrt{3} (2 \sqrt{5}) - \sqrt{3} \sqrt{3}$

Multiply $4$ by $5$ .

$20 + 2 \sqrt{5} \sqrt{3} - \sqrt{3} (2 \sqrt{5}) - \sqrt{3} \sqrt{3}$

Multiply $2 \sqrt{5} \sqrt{3}$ .

Combine using the product rule for radicals.

$20 + 2 \sqrt{3 \cdot 5} - \sqrt{3} (2 \sqrt{5}) - \sqrt{3} \sqrt{3}$

Multiply $3$ by $5$ .

$20 + 2 \sqrt{15} - \sqrt{3} (2 \sqrt{5}) - \sqrt{3} \sqrt{3}$

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

Multiply $2$ by $- 1$ .

$20 + 2 \sqrt{15} - 2 \sqrt{3} \sqrt{5} - \sqrt{3} \sqrt{3}$

Combine using the product rule for radicals.

$20 + 2 \sqrt{15} - 2 \sqrt{5 \cdot 3} - \sqrt{3} \sqrt{3}$

Multiply $5$ by $3$ .

$20 + 2 \sqrt{15} - 2 \sqrt{15} - \sqrt{3} \sqrt{3}$

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

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

$20 + 2 \sqrt{15} - 2 \sqrt{15} - ({\sqrt{3}}^{1} \sqrt{3})$

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

$20 + 2 \sqrt{15} - 2 \sqrt{15} - ({\sqrt{3}}^{1} {\sqrt{3}}^{1})$

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

$20 + 2 \sqrt{15} - 2 \sqrt{15} - {\sqrt{3}}^{1 + 1}$

Add $1$ and $1$ .

$20 + 2 \sqrt{15} - 2 \sqrt{15} - {\sqrt{3}}^{2}$

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

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

$20 + 2 \sqrt{15} - 2 \sqrt{15} - {(3^{\frac{1}{2}})}^{2}$

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

$20 + 2 \sqrt{15} - 2 \sqrt{15} - 3^{\frac{1}{2} \cdot 2}$

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

$20 + 2 \sqrt{15} - 2 \sqrt{15} - 3^{\frac{2}{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$20 + 2 \sqrt{15} - 2 \sqrt{15} - 3^{\frac{2}{2}}$

Rewrite the expression.

$20 + 2 \sqrt{15} - 2 \sqrt{15} - 3^{1}$

Evaluate the exponent.

$20 + 2 \sqrt{15} - 2 \sqrt{15} - 1 \cdot 3$

Multiply $- 1$ by $3$ .

$20 + 2 \sqrt{15} - 2 \sqrt{15} - 3$

Subtract $3$ from $20$ .

$17 + 2 \sqrt{15} - 2 \sqrt{15}$

Subtract $2 \sqrt{15}$ from $2 \sqrt{15}$ .

$17 + 0$

Add $17$ and $0$ .

$17$

Evaluate ( square root of 20- square root of 3)( square root of 20+ square root of 3)

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