Simplify the Radical Expression (5- square root of 2)(5+ square root of 2)

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

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

$5 \cdot 5 + 5 \sqrt{2} - \sqrt{2} \cdot 5 - \sqrt{2} \sqrt{2}$

Simplify and combine like terms.

Simplify each term.

Multiply $5$ by $5$ .

$25 + 5 \sqrt{2} - \sqrt{2} \cdot 5 - \sqrt{2} \sqrt{2}$

Multiply $5$ by $- 1$ .

$25 + 5 \sqrt{2} - 5 \sqrt{2} - \sqrt{2} \sqrt{2}$

Multiply $- \sqrt{2} \sqrt{2}$ .

Raise $\sqrt{2}$ to the power of $1$ .

$25 + 5 \sqrt{2} - 5 \sqrt{2} - ({\sqrt{2}}^{1} \sqrt{2})$

Raise $\sqrt{2}$ to the power of $1$ .

$25 + 5 \sqrt{2} - 5 \sqrt{2} - ({\sqrt{2}}^{1} {\sqrt{2}}^{1})$

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

$25 + 5 \sqrt{2} - 5 \sqrt{2} - {\sqrt{2}}^{1 + 1}$

Add $1$ and $1$ .

$25 + 5 \sqrt{2} - 5 \sqrt{2} - {\sqrt{2}}^{2}$

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

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

$25 + 5 \sqrt{2} - 5 \sqrt{2} - {(2^{\frac{1}{2}})}^{2}$

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

$25 + 5 \sqrt{2} - 5 \sqrt{2} - 2^{\frac{1}{2} \cdot 2}$

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

$25 + 5 \sqrt{2} - 5 \sqrt{2} - 2^{\frac{2}{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$25 + 5 \sqrt{2} - 5 \sqrt{2} - 2^{\frac{2}{2}}$

Rewrite the expression.

$25 + 5 \sqrt{2} - 5 \sqrt{2} - 2^{1}$

Evaluate the exponent.

$25 + 5 \sqrt{2} - 5 \sqrt{2} - 1 \cdot 2$

Multiply $- 1$ by $2$ .

$25 + 5 \sqrt{2} - 5 \sqrt{2} - 2$

Subtract $2$ from $25$ .

$23 + 5 \sqrt{2} - 5 \sqrt{2}$

Subtract $5 \sqrt{2}$ from $5 \sqrt{2}$ .

$23 + 0$

Add $23$ and $0$ .

$23$