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

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

Step 1

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Step 2

Simplify and combine like terms.

Simplify each term.

Multiply $3$ by $3$ .

$9 + 3 (- \sqrt{5}) + \sqrt{5} \cdot 3 + \sqrt{5} (- \sqrt{5})$

Multiply $- 1$ by $3$ .

$9 - 3 \sqrt{5} + \sqrt{5} \cdot 3 + \sqrt{5} (- \sqrt{5})$

Move $3$ to the left of $\sqrt{5}$ .

$9 - 3 \sqrt{5} + 3 \cdot \sqrt{5} + \sqrt{5} (- \sqrt{5})$

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

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

$9 - 3 \sqrt{5} + 3 \sqrt{5} - ({\sqrt{5}}^{1} \sqrt{5})$

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

$9 - 3 \sqrt{5} + 3 \sqrt{5} - ({\sqrt{5}}^{1} {\sqrt{5}}^{1})$

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

$9 - 3 \sqrt{5} + 3 \sqrt{5} - {\sqrt{5}}^{1 + 1}$

Add $1$ and $1$ .

$9 - 3 \sqrt{5} + 3 \sqrt{5} - {\sqrt{5}}^{2}$

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

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

$9 - 3 \sqrt{5} + 3 \sqrt{5} - {(5^{\frac{1}{2}})}^{2}$

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

$9 - 3 \sqrt{5} + 3 \sqrt{5} - 5^{\frac{1}{2} \cdot 2}$

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

$9 - 3 \sqrt{5} + 3 \sqrt{5} - 5^{\frac{2}{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$9 - 3 \sqrt{5} + 3 \sqrt{5} - 5^{\frac{2}{2}}$

Rewrite the expression.

$9 - 3 \sqrt{5} + 3 \sqrt{5} - 5^{1}$

Evaluate the exponent.

$9 - 3 \sqrt{5} + 3 \sqrt{5} - 1 \cdot 5$

Multiply $- 1$ by $5$ .

$9 - 3 \sqrt{5} + 3 \sqrt{5} - 5$

Subtract $5$ from $9$ .

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

Add $- 3 \sqrt{5}$ and $3 \sqrt{5}$ .

$4 + 0$

Add $4$ and $0$ .

$4$

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

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