Algebra Examples

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

Step 1

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

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Step 1.1

Apply the distributive property.

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

Step 1.2

Apply the distributive property.

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

Step 1.3

Apply the distributive property.

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

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.

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

Step 2.1.2

Multiply $3$ by $3$ .

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

Step 2.1.3

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

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

Step 2.1.4

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

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

Step 2.1.5

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

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Step 2.1.5.1

Combine using the product rule for radicals.

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

Step 2.1.5.2

Multiply $3$ by $5$ .

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

Step 2.1.6

Combine using the product rule for radicals.

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

Step 2.1.7

Multiply $5$ by $3$ .

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

Step 2.1.8

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

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Step 2.1.8.1

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

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

Step 2.1.8.2

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

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

Step 2.1.8.3

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

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

Step 2.1.8.4

Add $1$ and $1$ .

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

Step 2.1.9

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

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Step 2.1.9.1

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

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

Step 2.1.9.2

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

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

Step 2.1.9.3

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

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

Step 2.1.9.4

Cancel the common factor of $2$ .

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Step 2.1.9.4.1

Cancel the common factor.

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

Step 2.1.9.4.2

Rewrite the expression.

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

Step 2.1.9.5

Evaluate the exponent.

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

Step 2.1.10

Multiply $- 3$ by $5$ .

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

Step 2.2

Subtract $15$ from $3$ .

$- 12 - 3 \sqrt{15} + \sqrt{15}$

Step 2.3

Add $- 3 \sqrt{15}$ and $\sqrt{15}$ .

$- 12 - 2 \sqrt{15}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$- 12 - 2 \sqrt{15}$

Decimal Form:

$- 19.74596669 \dots$