To determine the absolute value of the complex number [tex]\( 5 - 4i \)[/tex], we use the formula for the modulus (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]
For the complex number [tex]\( 5 - 4i \)[/tex]:
- [tex]\( a = 5 \)[/tex]
- [tex]\( b = -4 \)[/tex]
Now, substitute these values into the formula:
[tex]\[ \left| 5 - 4i \right| = \sqrt{5^2 + (-4)^2} \][/tex]
[tex]\[ \left| 5 - 4i \right| = \sqrt{25 + 16} \][/tex]
[tex]\[ \left| 5 - 4i \right| = \sqrt{41} \][/tex]
Upon calculating the square root of 41, we obtain approximately:
[tex]\[ \sqrt{41} \approx 6.4031242374328485 \][/tex]
Given the options:
a. -6.4
b. 6
c. 6.04
d. 6.4
The closest and most appropriate answer is:
d. 6.4
