Solve frac{(sqrt{19}+{left(sqrt{5}right)}^2}{12+sqrt{95}}

$$\frac{ ( \sqrt{ 19 } + { \left( \sqrt{ 5 } \right) }^{ 2 } }{ 12+ \sqrt{ 95 } }$$

Evaluate

$$\frac{12\sqrt{19}+60-5\sqrt{95}-19\sqrt{5}}{49}\approx 0.430357633$$

Solution Steps

The square of \sqrt{5} is 5.

$$\frac{\sqrt{19}+5}{12+\sqrt{95}}$$

Rationalize the denominator of \frac{\sqrt{19}+5}{12+\sqrt{95}} by multiplying numerator and denominator by 12-\sqrt{95}.

$$\frac{\left(\sqrt{19}+5\right)\left(12-\sqrt{95}\right)}{\left(12+\sqrt{95}\right)\left(12-\sqrt{95}\right)}$$

Consider \left(12+\sqrt{95}\right)\left(12-\sqrt{95}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{\left(\sqrt{19}+5\right)\left(12-\sqrt{95}\right)}{12^{2}-\left(\sqrt{95}\right)^{2}}$$

Square 12. Square \sqrt{95}.

$$\frac{\left(\sqrt{19}+5\right)\left(12-\sqrt{95}\right)}{144-95}$$

Subtract 95 from 144 to get 49.

$$\frac{\left(\sqrt{19}+5\right)\left(12-\sqrt{95}\right)}{49}$$

Use the distributive property to multiply \sqrt{19}+5 by 12-\sqrt{95}.

$$\frac{12\sqrt{19}-\sqrt{19}\sqrt{95}+60-5\sqrt{95}}{49}$$

Factor 95=19\times 5. Rewrite the square root of the product \sqrt{19\times 5} as the product of square roots \sqrt{19}\sqrt{5}.

$$\frac{12\sqrt{19}-\sqrt{19}\sqrt{19}\sqrt{5}+60-5\sqrt{95}}{49}$$

Multiply \sqrt{19} and \sqrt{19} to get 19.

$$\frac{12\sqrt{19}-19\sqrt{5}+60-5\sqrt{95}}{49}$$