To find the absolute value of the complex number [tex]\(4 + 7i\)[/tex], we use the formula for the absolute value of a complex number [tex]\(a + bi\)[/tex], which is given by:
[tex]\[
\left| a + bi \right| = \sqrt{a^2 + b^2}
\][/tex]
In this case, the real part [tex]\(a\)[/tex] is 4 and the imaginary part [tex]\(b\)[/tex] is 7. Follow these steps:
1. Square the real part:
[tex]\[
4^2 = 16
\][/tex]
2. Square the imaginary part:
[tex]\[
7^2 = 49
\][/tex]
3. Sum the squares of the real and imaginary parts:
[tex]\[
16 + 49 = 65
\][/tex]
4. Take the square root of the sum to find the absolute value:
[tex]\[
\sqrt{65} \approx 8.06225774829855
\][/tex]
Thus, the absolute value of [tex]\(4 + 7i\)[/tex] is approximately [tex]\(8.06225774829855\)[/tex]. Therefore, the absolute value of [tex]\(4 + 7i\)[/tex] is equal to the square root of 65.
[tex]\[
\left| 4 + 7i \right| = \sqrt{65}
\][/tex]
