Simplify ( square root of 5+2 square root of 3)( square root of 5-2 square root of 3)

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

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Combine using the product rule for radicals.

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

Multiply $5$ by $5$ .

$\sqrt{25} + \sqrt{5} (- 2 \sqrt{3}) + 2 \sqrt{3} \sqrt{5} + 2 \sqrt{3} (- 2 \sqrt{3})$

Rewrite $25$ as $5^{2}$ .

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

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

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

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

Combine using the product rule for radicals.

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

Multiply $5$ by $3$ .

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

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

Combine using the product rule for radicals.

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

Multiply $5$ by $3$ .

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

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

Multiply $- 2$ by $2$ .

$5 - 2 \sqrt{15} + 2 \sqrt{15} - 4 \sqrt{3} \sqrt{3}$

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

$5 - 2 \sqrt{15} + 2 \sqrt{15} - 4 ({\sqrt{3}}^{1} \sqrt{3})$

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

$5 - 2 \sqrt{15} + 2 \sqrt{15} - 4 ({\sqrt{3}}^{1} {\sqrt{3}}^{1})$

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

$5 - 2 \sqrt{15} + 2 \sqrt{15} - 4 {\sqrt{3}}^{1 + 1}$

Add $1$ and $1$ .

$5 - 2 \sqrt{15} + 2 \sqrt{15} - 4 {\sqrt{3}}^{2}$

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

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

$5 - 2 \sqrt{15} + 2 \sqrt{15} - 4 {(3^{\frac{1}{2}})}^{2}$

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

$5 - 2 \sqrt{15} + 2 \sqrt{15} - 4 \cdot 3^{\frac{1}{2} \cdot 2}$

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

$5 - 2 \sqrt{15} + 2 \sqrt{15} - 4 \cdot 3^{\frac{2}{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$5 - 2 \sqrt{15} + 2 \sqrt{15} - 4 \cdot 3^{\frac{2}{2}}$

Rewrite the expression.

$5 - 2 \sqrt{15} + 2 \sqrt{15} - 4 \cdot 3^{1}$

Evaluate the exponent.

$5 - 2 \sqrt{15} + 2 \sqrt{15} - 4 \cdot 3$

Multiply $- 4$ by $3$ .

$5 - 2 \sqrt{15} + 2 \sqrt{15} - 12$

Subtract $12$ from $5$ .

$- 7 - 2 \sqrt{15} + 2 \sqrt{15}$

Add $- 2 \sqrt{15}$ and $2 \sqrt{15}$ .

$- 7 + 0$

Add $- 7$ and $0$ .

$- 7$

Simplify ( square root of 5+2 square root of 3)( square root of 5-2 square root of 3)

Related Questions

Support LCMGCF.com by disabling your adblocker.