Question

question 23 of 40
simplify the complex number. write the answer in the form a+bi. express numbers in ex
\\(\frac{4+\sqrt{-20}}{2}\\)
\\(\frac{4+\sqrt{-20}}{2} = \\)
question 24 of 40
translate the english phrase to an algebraic expression.
the difference of p and the cube root of q.

Explanation:

Question 23

Step1: Simplify the square root of a negative number

Recall that \(\sqrt{-n} = \sqrt{n} \cdot i\) for \(n > 0\). So, \(\sqrt{-20}=\sqrt{20}\cdot i\). And \(\sqrt{20}=\sqrt{4\times5} = 2\sqrt{5}\). Thus, \(\sqrt{-20}=2\sqrt{5}i\).

Step2: Substitute back into the expression and simplify

Substitute \(\sqrt{-20}=2\sqrt{5}i\) into \(\frac{4 + \sqrt{-20}}{2}\), we get \(\frac{4 + 2\sqrt{5}i}{2}\).
Now, divide each term in the numerator by 2: \(\frac{4}{2}+\frac{2\sqrt{5}i}{2}=2 + \sqrt{5}i\).

The phrase "the difference of \(p\) and the cube root of \(q\)" means we subtract the cube root of \(q\) from \(p\). The cube root of \(q\) is written as \(\sqrt[3]{q}\). So the algebraic expression is \(p-\sqrt[3]{q}\).

Answer:

\(2+\sqrt{5}i\)

Question 24