Solving sqrt(-8.3)sqrt(1 - i8): Tips and Tricks for Complex Numbers"

In summary, to solve sqrt(-8.3)sqrt(1 - i8), first simplify the expression so that there is only one square root. Then use the equation z = (a + bi)^2 and solve for a and b by equating the real and imaginary parts.
  • #1
naspek
181
0
how to solve sqrt(-8.3)sqrt(1 - i8)?


i try to solve it.. but got the wrong answer..

sqrt(-8.3)sqrt(1 - i8) = sqrt[(8.3i^2)(1 - 8i)]
= sqrt (8.3i^2 - 66.4i)
= 2.88i + 8.15

the answer should be.. 5.41 + i6.13
 
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  • #2
naspek said:
how to solve sqrt(-8.3)sqrt(1 - i8)?i try to solve it.. but got the wrong answer..

sqrt(-8.3)sqrt(1 - i8) = sqrt[(8.3i^2)(1 - 8i)]
= sqrt (8.3i^2 - 66.4i)
= 2.88i + 8.15
Here's a problem (in bold). You can't take the square root of a sum/difference separately. In other words,
[itex]\sqrt{a + b} \ne \sqrt{a} + \sqrt{b}[/itex]
 
  • #3
eumyang said:
Here's a problem (in bold). You can't take the square root of a sum/difference separately. In other words,
[itex]\sqrt{a + b} \ne \sqrt{a} + \sqrt{b}[/itex]

then... what should i do..? i got stuck there...
 
  • #4
Assuming you want just the principal square root, consider this: if there is a complex number a + bi such that
[itex]\sqrt{z} = a + bi[/itex],
then it makes sense that
[itex]z = (a + bi)^2[/itex].

First simplify the expression so that there is one square root. You sort of did that here (in bold):
naspek said:
sqrt(-8.3)sqrt(1 - i8) = sqrt[(8.3i^2)(1 - 8i)]
= sqrt (8.3i^2 - 66.4i)
= 2.88i + 8.15
... but there is a sign mistake. Also, forget about rewriting a negative as i2 in your 1st step.

Whatever is under the square root is your z. Take this:
[itex]z = (a + bi)^2[/itex]
and expand the right-hand side. Equate the real number parts and the imaginary number parts. You'll end up with 2 equations and 2 unknowns. Solve for a and b.
 
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FAQ: Solving sqrt(-8.3)sqrt(1 - i8): Tips and Tricks for Complex Numbers"

What are complex numbers and how are they used in solving equations?

Complex numbers are numbers that have both a real and an imaginary component. They are written in the form a + bi, where a represents the real part and bi represents the imaginary part. Complex numbers are used to solve equations that involve taking the square root of a negative number, which cannot be done with real numbers.

What is the process for solving sqrt(-8.3)sqrt(1 - i8)?

To solve this equation, we first need to simplify each square root separately. The square root of -8.3 can be written as 2.883i, and the square root of 1-i8 can be written as 2.828i. Then, we can use the rules of complex numbers to multiply the two simplified square roots together, giving us the final answer of 8.136i.

What are some tips for simplifying complex numbers?

One tip is to remember that the square root of -1 is represented by the letter i. This can help you identify the imaginary part of a complex number. Additionally, you can use the FOIL method to simplify the multiplication of two complex numbers, and remember that i^2 is equal to -1.

How do I know if my answer for sqrt(-8.3)sqrt(1 - i8) is correct?

You can check your answer by squaring it and seeing if you get the original equation. In this case, you would square 8.136i and see if it equals -8.3 + i8, which it does. This confirms that your answer is correct.

Can complex numbers be used in real-life applications?

Yes, complex numbers are used in many real-life applications, particularly in fields such as engineering, physics, and economics. They are also used in electronics, signal processing, and computer graphics. Complex numbers allow us to work with quantities that have both a real and imaginary component, making them useful in a wide range of applications.

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