Simplify (4 square root of 10+ square root of 11+ square root of 110+13)/( square root of 9+ square root of 10+ square root of 11)

$\frac{4 \sqrt{10} + \sqrt{11} + \sqrt{110} + 13}{\sqrt{9} + \sqrt{10} + \sqrt{11}}$

Step 1

Simplify the denominator.

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

$\frac{4 \sqrt{10} + \sqrt{11} + \sqrt{110} + 13}{\sqrt{3^{2}} + \sqrt{10} + \sqrt{11}}$

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

$\frac{4 \sqrt{10} + \sqrt{11} + \sqrt{110} + 13}{3 + \sqrt{10} + \sqrt{11}}$

Step 2

Multiply $\frac{4 \sqrt{10} + \sqrt{11} + \sqrt{110} + 13}{3 + \sqrt{10} + \sqrt{11}}$ by $\frac{3 + \sqrt{10} - \sqrt{11}}{3 + \sqrt{10} - \sqrt{11}}$ .

$\frac{4 \sqrt{10} + \sqrt{11} + \sqrt{110} + 13}{3 + \sqrt{10} + \sqrt{11}} \cdot \frac{3 + \sqrt{10} - \sqrt{11}}{3 + \sqrt{10} - \sqrt{11}}$

Step 3

Multiply $\frac{4 \sqrt{10} + \sqrt{11} + \sqrt{110} + 13}{3 + \sqrt{10} + \sqrt{11}}$ by $\frac{3 + \sqrt{10} - \sqrt{11}}{3 + \sqrt{10} - \sqrt{11}}$ .

$\frac{(4 \sqrt{10} + \sqrt{11} + \sqrt{110} + 13) (3 + \sqrt{10} - \sqrt{11})}{(3 + \sqrt{10} + \sqrt{11}) (3 + \sqrt{10} - \sqrt{11})}$

Step 4

Expand the denominator using the FOIL method.

$\frac{(4 \sqrt{10} + \sqrt{11} + \sqrt{110} + 13) (3 + \sqrt{10} - \sqrt{11})}{9 + 3 \sqrt{10} - 3 \sqrt{11} + \sqrt{10} \cdot 3 + {\sqrt{10}}^{2} - \sqrt{110} + \sqrt{11} \cdot 3 + \sqrt{110} - {\sqrt{11}}^{2}}$

Step 5

Simplify.

$\frac{(4 \sqrt{10} + \sqrt{11} + \sqrt{110} + 13) (3 + \sqrt{10} - \sqrt{11})}{8 + 6 \sqrt{10}}$

Step 6

Multiply $\frac{(4 \sqrt{10} + \sqrt{11} + \sqrt{110} + 13) (3 + \sqrt{10} - \sqrt{11})}{8 + 6 \sqrt{10}}$ by $\frac{8 - 6 \sqrt{10}}{8 - 6 \sqrt{10}}$ .

$\frac{(4 \sqrt{10} + \sqrt{11} + \sqrt{110} + 13) (3 + \sqrt{10} - \sqrt{11})}{8 + 6 \sqrt{10}} \cdot \frac{8 - 6 \sqrt{10}}{8 - 6 \sqrt{10}}$

Step 7

Multiply $\frac{(4 \sqrt{10} + \sqrt{11} + \sqrt{110} + 13) (3 + \sqrt{10} - \sqrt{11})}{8 + 6 \sqrt{10}}$ by $\frac{8 - 6 \sqrt{10}}{8 - 6 \sqrt{10}}$ .

$\frac{(4 \sqrt{10} + \sqrt{11} + \sqrt{110} + 13) (3 + \sqrt{10} - \sqrt{11}) (8 - 6 \sqrt{10})}{(8 + 6 \sqrt{10}) (8 - 6 \sqrt{10})}$

Step 8

Expand the denominator using the FOIL method.

$\frac{(4 \sqrt{10} + \sqrt{11} + \sqrt{110} + 13) (3 + \sqrt{10} - \sqrt{11}) (8 - 6 \sqrt{10})}{64 - 48 \sqrt{10} + 48 \sqrt{10} - 36 {\sqrt{10}}^{2}}$

Step 9

Simplify.

$\frac{(4 \sqrt{10} + \sqrt{11} + \sqrt{110} + 13) (3 + \sqrt{10} - \sqrt{11}) (8 - 6 \sqrt{10})}{- 296}$

Step 10

Cancel the common factor of $8 - 6 \sqrt{10}$ and $- 296$ .

Factor $2$ out of $(4 \sqrt{10} + \sqrt{11} + \sqrt{110} + 13) (3 + \sqrt{10} - \sqrt{11}) (8 - 6 \sqrt{10})$ .

$\frac{2 ((4 \sqrt{10} + \sqrt{11} + \sqrt{110} + 13) (3 + \sqrt{10} - \sqrt{11}) (4 - 3 \sqrt{10}))}{- 296}$

Cancel the common factors.

Factor $2$ out of $- 296$ .

$\frac{2 ((4 \sqrt{10} + \sqrt{11} + \sqrt{110} + 13) (3 + \sqrt{10} - \sqrt{11}) (4 - 3 \sqrt{10}))}{2 (- 148)}$

Cancel the common factor.

$\frac{2 ((4 \sqrt{10} + \sqrt{11} + \sqrt{110} + 13) (3 + \sqrt{10} - \sqrt{11}) (4 - 3 \sqrt{10}))}{2 \cdot - 148}$

Rewrite the expression.

$\frac{(4 \sqrt{10} + \sqrt{11} + \sqrt{110} + 13) (3 + \sqrt{10} - \sqrt{11}) (4 - 3 \sqrt{10})}{- 148}$

Step 11

Move the negative in front of the fraction.

$- \frac{(4 \sqrt{10} + \sqrt{11} + \sqrt{110} + 13) (3 + \sqrt{10} - \sqrt{11}) (4 - 3 \sqrt{10})}{148}$

Step 12

The result can be shown in multiple forms.

Exact Form:

$- \frac{(4 \sqrt{10} + \sqrt{11} + \sqrt{110} + 13) (3 + \sqrt{10} - \sqrt{11}) (4 - 3 \sqrt{10})}{148}$

Decimal Form:

$4.16227766 \dots$

Simplify (4 square root of 10+ square root of 11+ square root of 110+13)/( square root of 9+ square root of 10+ square root of 11)

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