Solve frac{(sqrt{17}+sqrt{7})^2}{12+sqrt{119}}

$$\frac { ( \sqrt { 17 } + \sqrt { 7 } ) ^ { 2 } } { 12 + \sqrt { 119 } }$$

Evaluate

$$2$$

Solution Steps

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(\sqrt{17}+\sqrt{7}\right)^{2}.

$$\frac{\left(\sqrt{17}\right)^{2}+2\sqrt{17}\sqrt{7}+\left(\sqrt{7}\right)^{2}}{12+\sqrt{119}}$$

The square of \sqrt{17} is 17.

$$\frac{17+2\sqrt{17}\sqrt{7}+\left(\sqrt{7}\right)^{2}}{12+\sqrt{119}}$$

To multiply \sqrt{17} and \sqrt{7}, multiply the numbers under the square root.

$$\frac{17+2\sqrt{119}+\left(\sqrt{7}\right)^{2}}{12+\sqrt{119}}$$

The square of \sqrt{7} is 7.

$$\frac{17+2\sqrt{119}+7}{12+\sqrt{119}}$$

Add 17 and 7 to get 24.

$$\frac{24+2\sqrt{119}}{12+\sqrt{119}}$$

Rationalize the denominator of \frac{24+2\sqrt{119}}{12+\sqrt{119}} by multiplying numerator and denominator by 12-\sqrt{119}.

$$\frac{\left(24+2\sqrt{119}\right)\left(12-\sqrt{119}\right)}{\left(12+\sqrt{119}\right)\left(12-\sqrt{119}\right)}$$

Consider \left(12+\sqrt{119}\right)\left(12-\sqrt{119}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{\left(24+2\sqrt{119}\right)\left(12-\sqrt{119}\right)}{12^{2}-\left(\sqrt{119}\right)^{2}}$$

Square 12. Square \sqrt{119}.

$$\frac{\left(24+2\sqrt{119}\right)\left(12-\sqrt{119}\right)}{144-119}$$

Subtract 119 from 144 to get 25.

$$\frac{\left(24+2\sqrt{119}\right)\left(12-\sqrt{119}\right)}{25}$$

Use the distributive property to multiply 24+2\sqrt{119} by 12-\sqrt{119} and combine like terms.

$$\frac{288-2\left(\sqrt{119}\right)^{2}}{25}$$

The square of \sqrt{119} is 119.

$$\frac{288-2\times 119}{25}$$

Multiply -2 and 119 to get -238.

$$\frac{288-238}{25}$$

Subtract 238 from 288 to get 50.

$$\frac{50}{25}$$

Divide 50 by 25 to get 2.

$$2$$

Factor

$$2$$