To find the modulus of a complex number, we use the formula:
[tex]\[ |a + bi| = \sqrt{a^2 + b^2} \][/tex]
where [tex]\( a \)[/tex] and [tex]\( b \)[/tex] are the real and imaginary parts of the complex number, respectively.
In this case, the given complex number is [tex]\( 5 + 3i \)[/tex].
Step-by-step solution:
1. Identify the real part [tex]\( a \)[/tex]:
[tex]\[ a = 5 \][/tex]
2. Identify the imaginary part [tex]\( b \)[/tex]:
[tex]\[ b = 3 \][/tex]
3. Square both [tex]\( a \)[/tex] and [tex]\( b \)[/tex]:
[tex]\[ a^2 = 5^2 = 25 \][/tex]
[tex]\[ b^2 = 3^2 = 9 \][/tex]
4. Add the squares:
[tex]\[ a^2 + b^2 = 25 + 9 = 34 \][/tex]
5. Take the square root of the sum:
[tex]\[ \sqrt{34} \][/tex]
Thus, the modulus of the complex number [tex]\( 5 + 3i \)[/tex] is:
[tex]\[ \sqrt{34} \][/tex]
Comparing this with the available options, the correct answer is:
[tex]\[ \boxed{\sqrt{34}} \][/tex]
