To find the absolute value (or magnitude) of a complex number [tex]\(7 + 12i\)[/tex], we use the formula for the magnitude of a complex number [tex]\(a + bi\)[/tex], which is given by:
[tex]\[
|a + bi| = \sqrt{a^2 + b^2}
\][/tex]
Here, [tex]\(a = 7\)[/tex] and [tex]\(b = 12\)[/tex]. Let's plug these values into the formula:
[tex]\[
|7 + 12i| = \sqrt{7^2 + 12^2}
\][/tex]
First, we need to calculate [tex]\(7^2\)[/tex] and [tex]\(12^2\)[/tex]:
[tex]\[
7^2 = 49
\][/tex]
[tex]\[
12^2 = 144
\][/tex]
Next, we add these results together:
[tex]\[
7^2 + 12^2 = 49 + 144 = 193
\][/tex]
Now, we take the square root of 193:
[tex]\[
\sqrt{193} \approx 13.892443989449804
\][/tex]
Hence, the absolute value of [tex]\(7 + 12i\)[/tex] is [tex]\(\sqrt{193}\)[/tex].
Therefore, the correct answer is:
[tex]\[
\sqrt{193}
\][/tex]
