Question

which of the following can be used to find the absolute value of 4 - 7i?
√(4² + (-7)²)
√(4²+(7i)²)
√(4² + 7²)
√((4 - 7i)²)

Explanation:

Step1: Recall absolute - value formula for complex numbers

For a complex number \(z = a+bi\), the absolute value \(|z|\) is given by \(|z|=\sqrt{a^{2}+b^{2}}\).

Step2: Identify \(a\) and \(b\) for \(z = 4 - 7i\)

For the complex number \(z = 4-7i\), \(a = 4\) and \(b=- 7\).

Step3: Calculate the absolute value

Substitute \(a = 4\) and \(b=-7\) into the formula \(|z|=\sqrt{a^{2}+b^{2}}\), we get \(|4 - 7i|=\sqrt{4^{2}+(-7)^{2}}\).

Answer:

\(\sqrt{4^{2}+(-7)^{2}}\) (the first option)