Algebra Examples

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

Step 1

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

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Step 1.1

Apply the distributive property.

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

Step 1.2

Apply the distributive property.

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

Step 1.3

Apply the distributive property.

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

Step 2

Simplify and combine like terms.

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Step 2.1

Simplify each term.

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Step 2.1.1

Combine using the product rule for radicals.

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

Step 2.1.2

Multiply $3$ by $3$ .

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

Step 2.1.3

Rewrite $9$ as $3^{2}$ .

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

Step 2.1.4

Pull terms out from under the radical.

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

Step 2.1.5

The absolute value is the distance between a number and zero. The distance between $0$ and $3$ is $3$ .

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

Step 2.1.6

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

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Step 2.1.6.1

Combine using the product rule for radicals.

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

Step 2.1.6.2

Multiply $3$ by $5$ .

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

Step 2.1.7

Combine using the product rule for radicals.

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

Step 2.1.8

Multiply $5$ by $3$ .

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

Step 2.1.9

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

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Step 2.1.9.1

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

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

Step 2.1.9.2

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

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

Step 2.1.9.3

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

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

Step 2.1.9.4

Add $1$ and $1$ .

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

Step 2.1.10

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

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Step 2.1.10.1

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

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

Step 2.1.10.2

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

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

Step 2.1.10.3

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

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

Step 2.1.10.4

Cancel the common factor of $2$ .

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Step 2.1.10.4.1

Cancel the common factor.

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

Step 2.1.10.4.2

Rewrite the expression.

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

Step 2.1.10.5

Evaluate the exponent.

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

Step 2.1.11

Multiply $- 1$ by $5$ .

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

Step 2.2

Subtract $5$ from $3$ .

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

Step 2.3

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

$- 2 + 0$

Step 2.4

Add $- 2$ and $0$ .

$- 2$