Certainly! Let's analyze the given complex numbers and find their absolute values to determine which one has an absolute value of 5.
A complex number is given in the form [tex]\( a + bi \)[/tex], where [tex]\( a \)[/tex] is the real part and [tex]\( b \)[/tex] is the imaginary part. The absolute value (or modulus) of a complex number [tex]\( a + bi \)[/tex] is calculated using the formula:
[tex]\[ |a + bi| = \sqrt{a^2 + b^2} \][/tex]
Now, let's find the absolute values of the given complex numbers:
1. For the complex number [tex]\(-3 + 4i\)[/tex]:
[tex]\[
| -3 + 4i | = \sqrt{(-3)^2 + (4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5
\][/tex]
2. For the complex number [tex]\(2 + 3i\)[/tex]:
[tex]\[
| 2 + 3i | = \sqrt{(2)^2 + (3)^2} = \sqrt{4 + 9} = \sqrt{13} \approx 3.605
\][/tex]
3. For the complex number [tex]\(7 - 2i\)[/tex]:
[tex]\[
| 7 - 2i | = \sqrt{(7)^2 + (-2)^2} = \sqrt{49 + 4} = \sqrt{53} \approx 7.280
\][/tex]
4. For the complex number [tex]\(9 + 4i\)[/tex]:
[tex]\[
| 9 + 4i | = \sqrt{(9)^2 + (4)^2} = \sqrt{81 + 16} = \sqrt{97} \approx 9.849
\][/tex]
Among the given complex numbers, we see that the absolute value of the complex number [tex]\(-3 + 4i\)[/tex] is 5.
Therefore, the complex number [tex]\(-3 + 4i\)[/tex] has an absolute value of 5.
