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Answer :
Sure! Let's break down where Petey went wrong and explain the correct steps to find the magnitude of the complex number [tex]\(5 + 13i\)[/tex].
### Petey's Mistake:
1. Petey calculated [tex]\((13i)^2\)[/tex] and got [tex]\((13i)^2 = 169 \cdot (-1)\)[/tex] correctly.
2. However, after calculating [tex]\((13i)^2 = 169(-1) = -169\)[/tex], he then mistakenly added 25 and [tex]\(-169\)[/tex] together. But the main issue was considering [tex]\(|5 + 13i|\)[/tex] should somehow involve adding an imaginary unit at the end, resulting in [tex]\(12i\)[/tex], which is incorrect.
### The Correct Approach:
To find the magnitude [tex]\(|5 + 13i|\)[/tex] of a complex number [tex]\(a + bi\)[/tex], you use the formula:
[tex]\[
|a + bi| = \sqrt{a^2 + b^2}
\][/tex]
Here, [tex]\(a = 5\)[/tex] and [tex]\(b = 13\)[/tex].
1. Calculate [tex]\(a^2\)[/tex]:
[tex]\[
5^2 = 25
\][/tex]
2. Calculate [tex]\(b^2\)[/tex]:
[tex]\[
13^2 = 169
\][/tex]
3. Add these together to find [tex]\(|5+13i|^2\)[/tex]:
[tex]\[
25 + 169 = 194
\][/tex]
4. Take the square root to find the magnitude:
[tex]\[
|5+13i| = \sqrt{194} \approx 13.93
\][/tex]
### Summary:
Petey's main error was in the interpretation of the squared terms and then incorrectly adding an imaginary unit. The correct magnitude of [tex]\(5 + 13i\)[/tex] is about 13.93.
### Petey's Mistake:
1. Petey calculated [tex]\((13i)^2\)[/tex] and got [tex]\((13i)^2 = 169 \cdot (-1)\)[/tex] correctly.
2. However, after calculating [tex]\((13i)^2 = 169(-1) = -169\)[/tex], he then mistakenly added 25 and [tex]\(-169\)[/tex] together. But the main issue was considering [tex]\(|5 + 13i|\)[/tex] should somehow involve adding an imaginary unit at the end, resulting in [tex]\(12i\)[/tex], which is incorrect.
### The Correct Approach:
To find the magnitude [tex]\(|5 + 13i|\)[/tex] of a complex number [tex]\(a + bi\)[/tex], you use the formula:
[tex]\[
|a + bi| = \sqrt{a^2 + b^2}
\][/tex]
Here, [tex]\(a = 5\)[/tex] and [tex]\(b = 13\)[/tex].
1. Calculate [tex]\(a^2\)[/tex]:
[tex]\[
5^2 = 25
\][/tex]
2. Calculate [tex]\(b^2\)[/tex]:
[tex]\[
13^2 = 169
\][/tex]
3. Add these together to find [tex]\(|5+13i|^2\)[/tex]:
[tex]\[
25 + 169 = 194
\][/tex]
4. Take the square root to find the magnitude:
[tex]\[
|5+13i| = \sqrt{194} \approx 13.93
\][/tex]
### Summary:
Petey's main error was in the interpretation of the squared terms and then incorrectly adding an imaginary unit. The correct magnitude of [tex]\(5 + 13i\)[/tex] is about 13.93.
Thanks for taking the time to read 3 Petey calculated tex 5 13i tex by finding begin array l 5 13i 2 5 2 13i 2 25 169 1 25 169 144. We hope the insights shared have been valuable and enhanced your understanding of the topic. Don�t hesitate to browse our website for more informative and engaging content!
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