Solve left(sqrt{5}+2right)left(sqrt{5}-2right)-sqrt{12}+-2^0+{left(1/3right)}^-1

$$\left( \sqrt{ 5 } +2 \right) \left( \sqrt{ 5 } -2 \right) - \sqrt{ 12 } + { -2 }^{ 0 } + { \left( \frac{ 1 }{ 3 } \right) }^{ -1 }$$

Evaluate

$$5-2\sqrt{3}\approx 1.535898385$$

Solution Steps

Consider \left(\sqrt{5}+2\right)\left(\sqrt{5}-2\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. Square 2.

$$\left(\sqrt{5}\right)^{2}-4-\sqrt{12}+\left(-2\right)^{0}+\left(\frac{1}{3}\right)^{-1}$$

The square of \sqrt{5} is 5.

$$5-4-\sqrt{12}+\left(-2\right)^{0}+\left(\frac{1}{3}\right)^{-1}$$

Subtract 4 from 5 to get 1.

$$1-\sqrt{12}+\left(-2\right)^{0}+\left(\frac{1}{3}\right)^{-1}$$

Factor 12=2^{2}\times 3. Rewrite the square root of the product \sqrt{2^{2}\times 3} as the product of square roots \sqrt{2^{2}}\sqrt{3}. Take the square root of 2^{2}.

$$1-2\sqrt{3}+\left(-2\right)^{0}+\left(\frac{1}{3}\right)^{-1}$$

Calculate -2 to the power of 0 and get 1.

$$1-2\sqrt{3}+1+\left(\frac{1}{3}\right)^{-1}$$

Add 1 and 1 to get 2.

$$2-2\sqrt{3}+\left(\frac{1}{3}\right)^{-1}$$

Calculate \frac{1}{3} to the power of -1 and get 3.

$$2-2\sqrt{3}+3$$

Add 2 and 3 to get 5.

$$5-2\sqrt{3}$$