Basic Math Examples

$\frac{(3 - \sqrt{7}) (11 + 4 \sqrt{7})}{5 + \sqrt{7}}$

Step 1

Expand $(3 - \sqrt{7}) (11 + 4 \sqrt{7})$ using the FOIL Method.

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Step 1.1

Apply the distributive property.

$\frac{3 (11 + 4 \sqrt{7}) - \sqrt{7} (11 + 4 \sqrt{7})}{5 + \sqrt{7}}$

Step 1.2

Apply the distributive property.

$\frac{3 \cdot 11 + 3 (4 \sqrt{7}) - \sqrt{7} (11 + 4 \sqrt{7})}{5 + \sqrt{7}}$

Step 1.3

Apply the distributive property.

$\frac{3 \cdot 11 + 3 (4 \sqrt{7}) - \sqrt{7} \cdot 11 - \sqrt{7} (4 \sqrt{7})}{5 + \sqrt{7}}$

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

Multiply $3$ by $11$ .

$\frac{33 + 3 (4 \sqrt{7}) - \sqrt{7} \cdot 11 - \sqrt{7} (4 \sqrt{7})}{5 + \sqrt{7}}$

Step 2.1.2

Multiply $4$ by $3$ .

$\frac{33 + 12 \sqrt{7} - \sqrt{7} \cdot 11 - \sqrt{7} (4 \sqrt{7})}{5 + \sqrt{7}}$

Step 2.1.3

Multiply $11$ by $- 1$ .

$\frac{33 + 12 \sqrt{7} - 11 \sqrt{7} - \sqrt{7} (4 \sqrt{7})}{5 + \sqrt{7}}$

Step 2.1.4

Multiply $- \sqrt{7} (4 \sqrt{7})$ .

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Step 2.1.4.1

Multiply $4$ by $- 1$ .

$\frac{33 + 12 \sqrt{7} - 11 \sqrt{7} - 4 \sqrt{7} \sqrt{7}}{5 + \sqrt{7}}$

Step 2.1.4.2

Raise $\sqrt{7}$ to the power of $1$ .

$\frac{33 + 12 \sqrt{7} - 11 \sqrt{7} - 4 ({\sqrt{7}}^{1} \sqrt{7})}{5 + \sqrt{7}}$

Step 2.1.4.3

Raise $\sqrt{7}$ to the power of $1$ .

$\frac{33 + 12 \sqrt{7} - 11 \sqrt{7} - 4 ({\sqrt{7}}^{1} {\sqrt{7}}^{1})}{5 + \sqrt{7}}$

Step 2.1.4.4

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

$\frac{33 + 12 \sqrt{7} - 11 \sqrt{7} - 4 {\sqrt{7}}^{1 + 1}}{5 + \sqrt{7}}$

Step 2.1.4.5

Add $1$ and $1$ .

$\frac{33 + 12 \sqrt{7} - 11 \sqrt{7} - 4 {\sqrt{7}}^{2}}{5 + \sqrt{7}}$

Step 2.1.5

Rewrite ${\sqrt{7}}^{2}$ as $7$ .

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Step 2.1.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{7}$ as $7^{\frac{1}{2}}$ .

$\frac{33 + 12 \sqrt{7} - 11 \sqrt{7} - 4 {(7^{\frac{1}{2}})}^{2}}{5 + \sqrt{7}}$

Step 2.1.5.2

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

$\frac{33 + 12 \sqrt{7} - 11 \sqrt{7} - 4 \cdot 7^{\frac{1}{2} \cdot 2}}{5 + \sqrt{7}}$

Step 2.1.5.3

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

$\frac{33 + 12 \sqrt{7} - 11 \sqrt{7} - 4 \cdot 7^{\frac{2}{2}}}{5 + \sqrt{7}}$

Step 2.1.5.4

Cancel the common factor of $2$ .

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Step 2.1.5.4.1

Cancel the common factor.

$\frac{33 + 12 \sqrt{7} - 11 \sqrt{7} - 4 \cdot 7^{\frac{2}{2}}}{5 + \sqrt{7}}$

Step 2.1.5.4.2

Rewrite the expression.

$\frac{33 + 12 \sqrt{7} - 11 \sqrt{7} - 4 \cdot 7^{1}}{5 + \sqrt{7}}$

Step 2.1.5.5

Evaluate the exponent.

$\frac{33 + 12 \sqrt{7} - 11 \sqrt{7} - 4 \cdot 7}{5 + \sqrt{7}}$

Step 2.1.6

Multiply $- 4$ by $7$ .

$\frac{33 + 12 \sqrt{7} - 11 \sqrt{7} - 28}{5 + \sqrt{7}}$

Step 2.2

Subtract $28$ from $33$ .

$\frac{5 + 12 \sqrt{7} - 11 \sqrt{7}}{5 + \sqrt{7}}$

Step 2.3

Subtract $11 \sqrt{7}$ from $12 \sqrt{7}$ .

$\frac{5 + \sqrt{7}}{5 + \sqrt{7}}$

Step 3

Cancel the common factor of $5 + \sqrt{7}$ .

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Step 3.1

Cancel the common factor.

$\frac{5 + \sqrt{7}}{5 + \sqrt{7}}$

Step 3.2

Rewrite the expression.

$1$