Basic Math Examples

$(\sqrt{11} + \sqrt{3}) (\sqrt{5} - \sqrt{3})$

Step 1

Expand $(\sqrt{11} + \sqrt{3}) (\sqrt{5} - \sqrt{3})$ using the FOIL Method.

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Step 1.1

Apply the distributive property.

$\sqrt{11} (\sqrt{5} - \sqrt{3}) + \sqrt{3} (\sqrt{5} - \sqrt{3})$

Step 1.2

Apply the distributive property.

$\sqrt{11} \sqrt{5} + \sqrt{11} (- \sqrt{3}) + \sqrt{3} (\sqrt{5} - \sqrt{3})$

Step 1.3

Apply the distributive property.

$\sqrt{11} \sqrt{5} + \sqrt{11} (- \sqrt{3}) + \sqrt{3} \sqrt{5} + \sqrt{3} (- \sqrt{3})$

Step 2

Simplify each term.

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Step 2.1

Combine using the product rule for radicals.

$\sqrt{11 \cdot 5} + \sqrt{11} (- \sqrt{3}) + \sqrt{3} \sqrt{5} + \sqrt{3} (- \sqrt{3})$

Step 2.2

Multiply $11$ by $5$ .

$\sqrt{55} + \sqrt{11} (- \sqrt{3}) + \sqrt{3} \sqrt{5} + \sqrt{3} (- \sqrt{3})$

Step 2.3

Multiply $\sqrt{11} (- \sqrt{3})$ .

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Step 2.3.1

Combine using the product rule for radicals.

$\sqrt{55} - \sqrt{11 \cdot 3} + \sqrt{3} \sqrt{5} + \sqrt{3} (- \sqrt{3})$

Step 2.3.2

Multiply $11$ by $3$ .

$\sqrt{55} - \sqrt{33} + \sqrt{3} \sqrt{5} + \sqrt{3} (- \sqrt{3})$

Step 2.4

Combine using the product rule for radicals.

$\sqrt{55} - \sqrt{33} + \sqrt{3 \cdot 5} + \sqrt{3} (- \sqrt{3})$

Step 2.5

Multiply $3$ by $5$ .

$\sqrt{55} - \sqrt{33} + \sqrt{15} + \sqrt{3} (- \sqrt{3})$

Step 2.6

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

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Step 2.6.1

Raise $\sqrt{3}$ to the power of $1$ .

$\sqrt{55} - \sqrt{33} + \sqrt{15} - ({\sqrt{3}}^{1} \sqrt{3})$

Step 2.6.2

Raise $\sqrt{3}$ to the power of $1$ .

$\sqrt{55} - \sqrt{33} + \sqrt{15} - ({\sqrt{3}}^{1} {\sqrt{3}}^{1})$

Step 2.6.3

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

$\sqrt{55} - \sqrt{33} + \sqrt{15} - {\sqrt{3}}^{1 + 1}$

Step 2.6.4

Add $1$ and $1$ .

$\sqrt{55} - \sqrt{33} + \sqrt{15} - {\sqrt{3}}^{2}$

Step 2.7

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 2.7.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$\sqrt{55} - \sqrt{33} + \sqrt{15} - {(3^{\frac{1}{2}})}^{2}$

Step 2.7.2

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

$\sqrt{55} - \sqrt{33} + \sqrt{15} - 3^{\frac{1}{2} \cdot 2}$

Step 2.7.3

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

$\sqrt{55} - \sqrt{33} + \sqrt{15} - 3^{\frac{2}{2}}$

Step 2.7.4

Cancel the common factor of $2$ .

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Step 2.7.4.1

Cancel the common factor.

$\sqrt{55} - \sqrt{33} + \sqrt{15} - 3^{\frac{2}{2}}$

Step 2.7.4.2

Rewrite the expression.

$\sqrt{55} - \sqrt{33} + \sqrt{15} - 3^{1}$

Step 2.7.5

Evaluate the exponent.

$\sqrt{55} - \sqrt{33} + \sqrt{15} - 1 \cdot 3$

Step 2.8

Multiply $- 1$ by $3$ .

$\sqrt{55} - \sqrt{33} + \sqrt{15} - 3$

Step 3

The result can be shown in multiple forms.

Exact Form:

$\sqrt{55} - \sqrt{33} + \sqrt{15} - 3$

Decimal Form:

$2.54461918 \dots$