Solve (-5)^0+sqrt[3]{-64}+(sqrt{2}+sqrt{3})(sqrt{2}-sqrt{3})+(1/2)^-1

$$( - 5 ) ^ { 0 } + \sqrt[ 3 ] { - 64 } + ( \sqrt { 2 } + \sqrt { 3 } ) ( \sqrt { 2 } - \sqrt { 3 } ) + ( \frac { 1 } { 2 } ) ^ { - 1 }$$

Evaluate

$$-2$$

Solution Steps

Calculate -5 to the power of 0 and get 1.

$$1+\sqrt[3]{-64}+\left(\sqrt{2}+\sqrt{3}\right)\left(\sqrt{2}-\sqrt{3}\right)+\left(\frac{1}{2}\right)^{-1}$$

Calculate \sqrt[3]{-64} and get -4.

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

Subtract 4 from 1 to get -3.

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

Consider \left(\sqrt{2}+\sqrt{3}\right)\left(\sqrt{2}-\sqrt{3}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

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

The square of \sqrt{2} is 2.

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

The square of \sqrt{3} is 3.

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

Subtract 3 from 2 to get -1.

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

Subtract 1 from -3 to get -4.

$$-4+\left(\frac{1}{2}\right)^{-1}$$

Calculate \frac{1}{2} to the power of -1 and get 2.

$$-4+2$$

Add -4 and 2 to get -2.

$$-2$$

Factor

$$-2$$