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

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

Step 1

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Step 2

Simplify and combine like terms.

Simplify each term.

Multiply $5$ by $5$ .

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

Multiply $2$ by $5$ .

$25 + 10 \sqrt{3} - 2 \sqrt{3} \cdot 5 - 2 \sqrt{3} (2 \sqrt{3})$

Multiply $5$ by $- 2$ .

$25 + 10 \sqrt{3} - 10 \sqrt{3} - 2 \sqrt{3} (2 \sqrt{3})$

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

Multiply $2$ by $- 2$ .

$25 + 10 \sqrt{3} - 10 \sqrt{3} - 4 \sqrt{3} \sqrt{3}$

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

$25 + 10 \sqrt{3} - 10 \sqrt{3} - 4 ({\sqrt{3}}^{1} \sqrt{3})$

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

$25 + 10 \sqrt{3} - 10 \sqrt{3} - 4 ({\sqrt{3}}^{1} {\sqrt{3}}^{1})$

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

$25 + 10 \sqrt{3} - 10 \sqrt{3} - 4 {\sqrt{3}}^{1 + 1}$

Add $1$ and $1$ .

$25 + 10 \sqrt{3} - 10 \sqrt{3} - 4 {\sqrt{3}}^{2}$

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

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

$25 + 10 \sqrt{3} - 10 \sqrt{3} - 4 {(3^{\frac{1}{2}})}^{2}$

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

$25 + 10 \sqrt{3} - 10 \sqrt{3} - 4 \cdot 3^{\frac{1}{2} \cdot 2}$

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

$25 + 10 \sqrt{3} - 10 \sqrt{3} - 4 \cdot 3^{\frac{2}{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$25 + 10 \sqrt{3} - 10 \sqrt{3} - 4 \cdot 3^{\frac{2}{2}}$

Rewrite the expression.

$25 + 10 \sqrt{3} - 10 \sqrt{3} - 4 \cdot 3^{1}$

Evaluate the exponent.

$25 + 10 \sqrt{3} - 10 \sqrt{3} - 4 \cdot 3$

Multiply $- 4$ by $3$ .

$25 + 10 \sqrt{3} - 10 \sqrt{3} - 12$

Subtract $12$ from $25$ .

$13 + 10 \sqrt{3} - 10 \sqrt{3}$

Subtract $10 \sqrt{3}$ from $10 \sqrt{3}$ .

$13 + 0$

Add $13$ and $0$ .

$13$

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

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