Calculus Examples

$\frac{20 - 7 \sqrt{7}}{11 - 4 \sqrt{7}}$

Step 1

Multiply $\frac{20 - 7 \sqrt{7}}{11 - 4 \sqrt{7}}$ by $\frac{11 + 4 \sqrt{7}}{11 + 4 \sqrt{7}}$ .

$\frac{20 - 7 \sqrt{7}}{11 - 4 \sqrt{7}} \cdot \frac{11 + 4 \sqrt{7}}{11 + 4 \sqrt{7}}$

Step 2

Combine fractions.

Tap for more steps...

Step 2.1

Multiply $\frac{20 - 7 \sqrt{7}}{11 - 4 \sqrt{7}}$ by $\frac{11 + 4 \sqrt{7}}{11 + 4 \sqrt{7}}$ .

$\frac{(20 - 7 \sqrt{7}) (11 + 4 \sqrt{7})}{(11 - 4 \sqrt{7}) (11 + 4 \sqrt{7})}$

Step 2.2

Expand the denominator using the FOIL method.

$\frac{(20 - 7 \sqrt{7}) (11 + 4 \sqrt{7})}{121 + 44 \sqrt{7} - 44 \sqrt{7} - 16 {\sqrt{7}}^{2}}$

Step 2.3

Simplify.

$\frac{(20 - 7 \sqrt{7}) (11 + 4 \sqrt{7})}{9}$

Step 3

Expand $(20 - 7 \sqrt{7}) (11 + 4 \sqrt{7})$ using the FOIL Method.

Tap for more steps...

Step 3.1

Apply the distributive property.

$\frac{20 (11 + 4 \sqrt{7}) - 7 \sqrt{7} (11 + 4 \sqrt{7})}{9}$

Step 3.2

Apply the distributive property.

$\frac{20 \cdot 11 + 20 (4 \sqrt{7}) - 7 \sqrt{7} (11 + 4 \sqrt{7})}{9}$

Step 3.3

Apply the distributive property.

$\frac{20 \cdot 11 + 20 (4 \sqrt{7}) - 7 \sqrt{7} \cdot 11 - 7 \sqrt{7} (4 \sqrt{7})}{9}$

Step 4

Simplify and combine like terms.

Tap for more steps...

Step 4.1

Simplify each term.

Tap for more steps...

Step 4.1.1

Multiply $20$ by $11$ .

$\frac{220 + 20 (4 \sqrt{7}) - 7 \sqrt{7} \cdot 11 - 7 \sqrt{7} (4 \sqrt{7})}{9}$

Step 4.1.2

Multiply $4$ by $20$ .

$\frac{220 + 80 \sqrt{7} - 7 \sqrt{7} \cdot 11 - 7 \sqrt{7} (4 \sqrt{7})}{9}$

Step 4.1.3

Multiply $11$ by $- 7$ .

$\frac{220 + 80 \sqrt{7} - 77 \sqrt{7} - 7 \sqrt{7} (4 \sqrt{7})}{9}$

Step 4.1.4

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

Tap for more steps...

Step 4.1.4.1

Multiply $4$ by $- 7$ .

$\frac{220 + 80 \sqrt{7} - 77 \sqrt{7} - 28 \sqrt{7} \sqrt{7}}{9}$

Step 4.1.4.2

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

$\frac{220 + 80 \sqrt{7} - 77 \sqrt{7} - 28 ({\sqrt{7}}^{1} \sqrt{7})}{9}$

Step 4.1.4.3

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

$\frac{220 + 80 \sqrt{7} - 77 \sqrt{7} - 28 ({\sqrt{7}}^{1} {\sqrt{7}}^{1})}{9}$

Step 4.1.4.4

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

$\frac{220 + 80 \sqrt{7} - 77 \sqrt{7} - 28 {\sqrt{7}}^{1 + 1}}{9}$

Step 4.1.4.5

Add $1$ and $1$ .

$\frac{220 + 80 \sqrt{7} - 77 \sqrt{7} - 28 {\sqrt{7}}^{2}}{9}$

Step 4.1.5

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

Tap for more steps...

Step 4.1.5.1

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

$\frac{220 + 80 \sqrt{7} - 77 \sqrt{7} - 28 {(7^{\frac{1}{2}})}^{2}}{9}$

Step 4.1.5.2

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

$\frac{220 + 80 \sqrt{7} - 77 \sqrt{7} - 28 \cdot 7^{\frac{1}{2} \cdot 2}}{9}$

Step 4.1.5.3

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

$\frac{220 + 80 \sqrt{7} - 77 \sqrt{7} - 28 \cdot 7^{\frac{2}{2}}}{9}$

Step 4.1.5.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 4.1.5.4.1

Cancel the common factor.

$\frac{220 + 80 \sqrt{7} - 77 \sqrt{7} - 28 \cdot 7^{\frac{2}{2}}}{9}$

Step 4.1.5.4.2

Rewrite the expression.

$\frac{220 + 80 \sqrt{7} - 77 \sqrt{7} - 28 \cdot 7^{1}}{9}$

Step 4.1.5.5

Evaluate the exponent.

$\frac{220 + 80 \sqrt{7} - 77 \sqrt{7} - 28 \cdot 7}{9}$

Step 4.1.6

Multiply $- 28$ by $7$ .

$\frac{220 + 80 \sqrt{7} - 77 \sqrt{7} - 196}{9}$

Step 4.2

Subtract $196$ from $220$ .

$\frac{24 + 80 \sqrt{7} - 77 \sqrt{7}}{9}$

Step 4.3

Subtract $77 \sqrt{7}$ from $80 \sqrt{7}$ .

$\frac{24 + 3 \sqrt{7}}{9}$

Step 5

Cancel the common factor of $24 + 3 \sqrt{7}$ and $9$ .

Tap for more steps...

Step 5.1

Factor $3$ out of $24$ .

$\frac{3 \cdot 8 + 3 \sqrt{7}}{9}$

Step 5.2

Factor $3$ out of $3 \sqrt{7}$ .

$\frac{3 \cdot 8 + 3 (\sqrt{7})}{9}$

Step 5.3

Factor $3$ out of $3 \cdot 8 + 3 (\sqrt{7})$ .

$\frac{3 \cdot (8 + \sqrt{7})}{9}$

Step 5.4

Cancel the common factors.

Tap for more steps...

Step 5.4.1

Factor $3$ out of $9$ .

$\frac{3 \cdot (8 + \sqrt{7})}{3 (3)}$

Step 5.4.2

Cancel the common factor.

$\frac{3 \cdot (8 + \sqrt{7})}{3 \cdot 3}$

Step 5.4.3

Rewrite the expression.

$\frac{8 + \sqrt{7}}{3}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\frac{8 + \sqrt{7}}{3}$

Decimal Form:

$3.54858377 \dots$