Basic Math Examples

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

Step 1

Combine using the product rule for radicals.

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

Step 2

Expand $(9 + 3 \sqrt{5}) (9 - 3 \sqrt{5})$ using the FOIL Method.

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Step 2.1

Apply the distributive property.

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

Step 2.2

Apply the distributive property.

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

Step 2.3

Apply the distributive property.

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

Step 3

Simplify and combine like terms.

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Step 3.1

Simplify each term.

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Step 3.1.1

Multiply $9$ by $9$ .

$\sqrt{81 + 9 (- 3 \sqrt{5}) + 3 \sqrt{5} \cdot 9 + 3 \sqrt{5} (- 3 \sqrt{5})}$

Step 3.1.2

Multiply $- 3$ by $9$ .

$\sqrt{81 - 27 \sqrt{5} + 3 \sqrt{5} \cdot 9 + 3 \sqrt{5} (- 3 \sqrt{5})}$

Step 3.1.3

Multiply $9$ by $3$ .

$\sqrt{81 - 27 \sqrt{5} + 27 \sqrt{5} + 3 \sqrt{5} (- 3 \sqrt{5})}$

Step 3.1.4

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

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Step 3.1.4.1

Multiply $- 3$ by $3$ .

$\sqrt{81 - 27 \sqrt{5} + 27 \sqrt{5} - 9 \sqrt{5} \sqrt{5}}$

Step 3.1.4.2

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

$\sqrt{81 - 27 \sqrt{5} + 27 \sqrt{5} - 9 ({\sqrt{5}}^{1} \sqrt{5})}$

Step 3.1.4.3

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

$\sqrt{81 - 27 \sqrt{5} + 27 \sqrt{5} - 9 ({\sqrt{5}}^{1} {\sqrt{5}}^{1})}$

Step 3.1.4.4

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

$\sqrt{81 - 27 \sqrt{5} + 27 \sqrt{5} - 9 {\sqrt{5}}^{1 + 1}}$

Step 3.1.4.5

Add $1$ and $1$ .

$\sqrt{81 - 27 \sqrt{5} + 27 \sqrt{5} - 9 {\sqrt{5}}^{2}}$

Step 3.1.5

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

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Step 3.1.5.1

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

$\sqrt{81 - 27 \sqrt{5} + 27 \sqrt{5} - 9 {(5^{\frac{1}{2}})}^{2}}$

Step 3.1.5.2

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

$\sqrt{81 - 27 \sqrt{5} + 27 \sqrt{5} - 9 \cdot 5^{\frac{1}{2} \cdot 2}}$

Step 3.1.5.3

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

$\sqrt{81 - 27 \sqrt{5} + 27 \sqrt{5} - 9 \cdot 5^{\frac{2}{2}}}$

Step 3.1.5.4

Cancel the common factor of $2$ .

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Step 3.1.5.4.1

Cancel the common factor.

$\sqrt{81 - 27 \sqrt{5} + 27 \sqrt{5} - 9 \cdot 5^{\frac{2}{2}}}$

Step 3.1.5.4.2

Rewrite the expression.

$\sqrt{81 - 27 \sqrt{5} + 27 \sqrt{5} - 9 \cdot 5^{1}}$

Step 3.1.5.5

Evaluate the exponent.

$\sqrt{81 - 27 \sqrt{5} + 27 \sqrt{5} - 9 \cdot 5}$

Step 3.1.6

Multiply $- 9$ by $5$ .

$\sqrt{81 - 27 \sqrt{5} + 27 \sqrt{5} - 45}$

Step 3.2

Subtract $45$ from $81$ .

$\sqrt{36 - 27 \sqrt{5} + 27 \sqrt{5}}$

Step 3.3

Add $- 27 \sqrt{5}$ and $27 \sqrt{5}$ .

$\sqrt{36 + 0}$

Step 3.4

Add $36$ and $0$ .

$\sqrt{36}$

Step 4

Rewrite $36$ as $6^{2}$ .

$\sqrt{6^{2}}$

Step 5

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

$6$