To find the absolute value of a complex number [tex]\( a + bi \)[/tex], we use the formula:
[tex]\[ |a + bi| = \sqrt{a^2 + b^2} \][/tex]
For the complex number [tex]\( 4 - 7i \)[/tex]:
1. Identify the real part [tex]\( a = 4 \)[/tex] and the imaginary part [tex]\( b = -7 \)[/tex].
2. Substitute [tex]\( a \)[/tex] and [tex]\( b \)[/tex] into the formula:
[tex]\[ |4 - 7i| = \sqrt{4^2 + (-7)^2} \][/tex]
So, the absolute value is:
[tex]\[ \sqrt{4^2 + (-7)^2} \][/tex]
Let's look at each of the given options:
1. [tex]\(\sqrt{4^2 + (-7)^2}\)[/tex] correctly represents the absolute value of [tex]\( 4 - 7i \)[/tex].
2. [tex]\(\sqrt{4^2 + (77)^2}\)[/tex] is not correct because it uses 77 instead of -7 for the imaginary part.
3. [tex]\(\sqrt{4^2 + 7^2}\)[/tex] incorrectly uses 7 instead of -7 for the imaginary part, which could change the result since the square of 7 is different from the square of -7.
4. [tex]\(\sqrt{(4 - 7i)^2}\)[/tex] does not correctly represent the absolute value formula, as squaring the entire complex number does not follow the absolute value calculation process.
Thus, the correct option to find the absolute value of [tex]\( 4 - 7i \)[/tex] is:
[tex]\[ \sqrt{4^2 + (-7)^2} \][/tex]
Therefore, the correct answer is the first option.
