Solve 2-sqrt{5}quadtext{(iii)}(3+sqrt{23})-sqrt{23}

$$2 - \sqrt { 5 } \quad \text { (iii) } ( 3 + \sqrt { 23 } ) - \sqrt { 23 }$$

Evaluate

$$2-\sqrt{23}+\sqrt{5}\left(\sqrt{23}+3\right)i\approx -2.795831523+17.432009227i$$

Solution Steps

Multiply i and i to get -1.

$$2-\sqrt{5}\left(-i\right)\left(3+\sqrt{23}\right)-\sqrt{23}$$

Use the distributive property to multiply \sqrt{5}\left(-i\right) by 3+\sqrt{23}.

$$2-\left(3\sqrt{5}\left(-i\right)-i\sqrt{5}\sqrt{23}\right)-\sqrt{23}$$

Multiply 3 and -i to get -3i.

$$2-\left(-3i\sqrt{5}-i\sqrt{5}\sqrt{23}\right)-\sqrt{23}$$

To multiply \sqrt{5} and \sqrt{23}, multiply the numbers under the square root.

$$2-\left(-3i\sqrt{5}-i\sqrt{115}\right)-\sqrt{23}$$

To find the opposite of -3i\sqrt{5}-i\sqrt{115}, find the opposite of each term.

$$2+3i\sqrt{5}+i\sqrt{115}-\sqrt{23}$$

Real Part

$$2-\sqrt{23}$$