Type The Correct Answer In Each Box.What Values Of $a$ And $b$ Make This Equation True?${ (4+\sqrt{-49})-2\left(\sqrt{(-4)^2}+\sqrt{-324}\right)=a+b I }$ { A=\square \} ${ B=\square }$

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Introduction

In this article, we will explore the values of aa and bb that make the given equation true. The equation involves complex numbers, which are numbers that have both real and imaginary parts. We will use the properties of complex numbers to simplify the equation and find the values of aa and bb.

The Given Equation

The given equation is:

(4+−49)−2((−4)2+−324)=a+bi(4+\sqrt{-49})-2\left(\sqrt{(-4)^2}+\sqrt{-324}\right)=a+b i

Simplifying the Equation

To simplify the equation, we will start by evaluating the expressions inside the square roots.

−49=−1⋅49=−1⋅49=7i\sqrt{-49} = \sqrt{-1 \cdot 49} = \sqrt{-1} \cdot \sqrt{49} = 7i

(−4)2=16=4\sqrt{(-4)^2} = \sqrt{16} = 4

−324=−1⋅324=−1⋅324=18i\sqrt{-324} = \sqrt{-1 \cdot 324} = \sqrt{-1} \cdot \sqrt{324} = 18i

Substituting the Simplified Expressions

Now, we will substitute the simplified expressions back into the original equation.

(4+7i)−2(4+18i)=a+bi(4+7i)-2(4+18i)=a+b i

Expanding and Simplifying

Next, we will expand and simplify the equation.

4+7i−8−36i=a+bi4+7i-8-36i=a+b i

Combining Like Terms

Now, we will combine like terms.

−4−29i=a+bi-4-29i=a+b i

Equating Real and Imaginary Parts

Since the equation is equal to a+bia+b i, we can equate the real and imaginary parts.

−4=a-4=a

−29=b-29=b

Conclusion

In conclusion, the values of aa and bb that make the equation true are a=−4a=-4 and b=−29b=-29.

Final Answer

The final answer is:

a=−4a=\boxed{-4}

b=−29b=\boxed{-29}

Discussion

This problem involves simplifying an equation with complex numbers and finding the values of aa and bb that make the equation true. The key steps in solving this problem are:

  1. Evaluating the expressions inside the square roots
  2. Substituting the simplified expressions back into the original equation
  3. Expanding and simplifying the equation
  4. Combining like terms
  5. Equating the real and imaginary parts

By following these steps, we can find the values of aa and bb that make the equation true.

Related Topics

  • Complex numbers
  • Simplifying equations with complex numbers
  • Equating real and imaginary parts

Further Reading

Introduction

In our previous article, we explored the values of aa and bb that make the given equation true. We simplified the equation, equated the real and imaginary parts, and found the values of aa and bb. In this article, we will answer some frequently asked questions related to the problem.

Q: What is the main concept behind solving this problem?

A: The main concept behind solving this problem is the properties of complex numbers. Complex numbers are numbers that have both real and imaginary parts. In this problem, we used the properties of complex numbers to simplify the equation and find the values of aa and bb.

Q: How do I simplify an equation with complex numbers?

A: To simplify an equation with complex numbers, you need to follow these steps:

  1. Evaluate the expressions inside the square roots.
  2. Substitute the simplified expressions back into the original equation.
  3. Expand and simplify the equation.
  4. Combine like terms.
  5. Equate the real and imaginary parts.

Q: What is the difference between real and imaginary parts?

A: The real part of a complex number is the part that is not multiplied by ii, while the imaginary part is the part that is multiplied by ii. In this problem, we equated the real and imaginary parts to find the values of aa and bb.

Q: How do I know when to use ii and when to use −i-i?

A: You use ii when the expression is multiplied by ii, and you use −i-i when the expression is multiplied by −i-i. In this problem, we used ii when we simplified the expression −49\sqrt{-49}.

Q: Can I use this method to solve other problems with complex numbers?

A: Yes, you can use this method to solve other problems with complex numbers. The steps we followed in this problem are general steps that can be applied to any problem with complex numbers.

Q: What are some common mistakes to avoid when solving problems with complex numbers?

A: Some common mistakes to avoid when solving problems with complex numbers are:

  • Not evaluating the expressions inside the square roots correctly
  • Not substituting the simplified expressions back into the original equation correctly
  • Not expanding and simplifying the equation correctly
  • Not combining like terms correctly
  • Not equating the real and imaginary parts correctly

Q: How can I practice solving problems with complex numbers?

A: You can practice solving problems with complex numbers by:

  • Working on practice problems
  • Using online resources, such as Khan Academy or Mathway
  • Asking your teacher or tutor for help
  • Joining a study group or online community to discuss math problems

Conclusion

In conclusion, solving problems with complex numbers requires a good understanding of the properties of complex numbers and the steps to simplify an equation. By following the steps we outlined in this article, you can solve problems with complex numbers and find the values of aa and bb that make the equation true.

Final Answer

The final answer is:

a=−4a=\boxed{-4}

b=−29b=\boxed{-29}

Related Topics

  • Complex numbers
  • Simplifying equations with complex numbers
  • Equating real and imaginary parts

Further Reading