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)
done

Explanation:

Step1: Recall the formula for the absolute value (modulus) of a complex number \( z = a + bi \), which is \( |z|=\sqrt{a^{2}+b^{2}} \).

Step2: Identify \( a \) and \( b \) for the complex number \( 4 - 7i \). Here, \( a = 4 \) and \( b=-7 \).

Step3: Substitute \( a \) and \( b \) into the formula. So \( |4 - 7i|=\sqrt{4^{2}+(-7)^{2}} \). We can also note that \( (-7)^{2}=7^{2} \), so \( \sqrt{4^{2}+(-7)^{2}}=\sqrt{4^{2}+7^{2}} \) as well. Let's check each option:

- First option: \( \sqrt{4^{2}+(-7)^{2}} \) matches the formula.
- Second option: \( \sqrt{4^{2}+(7i)^{2}} \) is incorrect because we square the real and imaginary parts (without the \( i \) for the imaginary part's square, and \( (7i)^{2}=-49 \), which is not what we need).
- Third option: \( \sqrt{4^{2}+7^{2}} \) is equivalent to the first option since \( (-7)^{2}=7^{2} \), so this is also correct. Wait, but let's check the original options again. Wait, the third option is \( \sqrt{4^{2}+7^{2}} \), the first is \( \sqrt{4^{2}+(-7)^{2}} \). But let's see the options. Wait, maybe the third option is also correct? Wait no, let's re - examine. Wait, the complex number is \( 4 - 7i \), so \( b=-7 \), so \( b^{2}=(-7)^{2}=49 \), and \( 7^{2}=49 \) too. So both \( \sqrt{4^{2}+(-7)^{2}} \) and \( \sqrt{4^{2}+7^{2}} \) are correct? But let's check the options. Wait the first option is \( \sqrt{4^{2}+(-7)^{2}} \), the third is \( \sqrt{4^{2}+7^{2}} \). Wait maybe the question has a typo or maybe I misread. Wait no, let's check the formula again. The modulus of \( a + bi \) is \( \sqrt{a^{2}+b^{2}} \), where \( b \) is the coefficient of \( i \). For \( 4-7i \), the coefficient of \( i \) is \( -7 \), so \( b = - 7 \), so \( b^{2}=(-7)^{2}=49 \), and \( 7^{2}=49 \), so \( \sqrt{4^{2}+(-7)^{2}}=\sqrt{4^{2}+7^{2}} \). But let's check the options:

First option: \( \sqrt{4^{2}+(-7)^{2}} \)

Second option: \( \sqrt{4^{2}+(7i)^{2}} \) (wrong, because we square the real and imaginary parts, the imaginary part is \( -7 \), not \( 7i \))

Third option: \( \sqrt{4^{2}+7^{2}} \) (correct as \( (-7)^2 = 7^2 \))

Fourth option: \( \sqrt{(4 - 7i)^{2}} \) (wrong, because \( |z|=\sqrt{z\overline{z}} \), and \( (4 - 7i)^{2}=16-56i + 49i^{2}=16-56i - 49=-33-56i \), and \( \sqrt{-33 - 56i} \) is not the modulus of \( 4 - 7i \))

Wait, but maybe the question considers that \( (-7)^2 = 7^2 \), so both first and third are correct? But let's check the original problem's options again. Wait the first option is \( \sqrt{4^2+(-7)^2} \), the third is \( \sqrt{4^2 + 7^2} \). But maybe in the context of the question, the first option is more direct. But also, \( (-7)^2=7^2 \), so \( \sqrt{4^2+(-7)^2}=\sqrt{4^2 + 7^2} \). So both first and third are correct? Wait no, let's check the formula again. The modulus of \( a+bi \) is \( \sqrt{a^2 + b^2} \), where \( b \) is the imaginary part. For \( 4-7i \), \( b = - 7 \), so \( b^2=(-7)^2 = 49 \), and \( 7^2 = 49 \), so \( \sqrt{4^2+(-7)^2}=\sqrt{4^2 + 7^2} \). So both the first and third options are correct? But let's check the options given. Wait the first option is \( \sqrt{4^2+(-7)^2} \), the third is \( \sqrt{4^2 + 7^2} \). Maybe the question expects the first option as it directly uses \( b=-7 \), or the third as \( (-7)^2 = 7^2 \). But let's see the options again. Wait the first option is \( \sqrt{4^2+(-7)^2} \), which is the direct application of the formula with \( b = - 7 \), and the third is \( \sqrt{4^2+7^2} \), which is equivalent. But let's check the options:

Looking at the options, the first option is \( \sqrt{4^2+(-7)^2} \), the third is \( \sqrt{4^2 + 7^2} \). Let's verify with the formula. The modulus of \( 4-7i \) is \( \sqrt{4^2+(-7)^2}=\sqrt{16 + 49}=\sqrt{65} \), and \( \sqrt{4^2+7^2}=\sqrt{16 + 49}=\sqrt{65} \), and \( \sqrt{4^2+(-7)^2} \) also gives \( \sqrt{65} \). The fourth option: \( \sqrt{(4 - 7i)^2}=\sqrt{16-56i + 49i^2}=\sqrt{16-56i - 49}=\sqrt{-33-56i} \), which is not a real number, while the modulus should be a real number. The second option: \( \sqrt{4^2+(7i)^2}=\sqrt{16 + 49i^2}=\sqrt{16-49}=\sqrt{-33} \), which is not real. So the correct options are the first and the third? But maybe in the question's context, the first option is the more direct application. Wait, but let's check the options again. The first option is \( \sqrt{4^2+(-7)^2} \), the third is \( \sqrt{4^2 + 7^2} \). Since \( (-7)^2 = 7^2 \), both are correct. But maybe the question considers the first option as the answer.

Answer:

A. \( \sqrt{4^{2}+(-7)^{2}} \) (and also C. \( \sqrt{4^{2}+7^{2}} \) is correct, but if we have to choose from the given options, the first option \( \sqrt{4^{2}+(-7)^{2}} \) is a direct application of the modulus formula for \( 4 - 7i \) with \( a = 4 \) and \( b=-7 \))