Simplify The Expression: 3 × ( − 1 + 4 I ) + ( 3 − 2 I 3 \times (-1 + 4i) + (3 - 2i 3 × ( − 1 + 4 I ) + ( 3 − 2 I ]

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Introduction

In mathematics, simplifying expressions is a crucial step in solving problems and understanding complex concepts. The given expression, 3×(−1+4i)+(3−2i)3 \times (-1 + 4i) + (3 - 2i), involves multiplication and addition of complex numbers. In this article, we will simplify the expression step by step, using the properties of complex numbers.

Understanding Complex Numbers

Complex numbers are mathematical expressions that consist of a real part and an imaginary part. The imaginary part is denoted by the letter ii, where i=−1i = \sqrt{-1}. Complex numbers can be added, subtracted, multiplied, and divided, just like real numbers. However, the rules for these operations are slightly different.

Properties of Complex Numbers

  • Addition: When adding complex numbers, we add the real parts and the imaginary parts separately.
  • Subtraction: When subtracting complex numbers, we subtract the real parts and the imaginary parts separately.
  • Multiplication: When multiplying complex numbers, we use the distributive property and the fact that i2=−1i^2 = -1.
  • Division: When dividing complex numbers, we multiply the numerator and denominator by the conjugate of the denominator.

Simplifying the Expression

To simplify the expression 3×(−1+4i)+(3−2i)3 \times (-1 + 4i) + (3 - 2i), we will follow the order of operations (PEMDAS):

  1. Multiply 33 by (−1+4i)(-1 + 4i) using the distributive property.
  2. Add the result to (3−2i)(3 - 2i).

Step 1: Multiply 33 by (−1+4i)(-1 + 4i)

Using the distributive property, we can multiply 33 by (−1+4i)(-1 + 4i) as follows:

3×(−1+4i)=3×(−1)+3×4i=−3+12i3 \times (-1 + 4i) = 3 \times (-1) + 3 \times 4i = -3 + 12i

Step 2: Add the Result to (3−2i)(3 - 2i)

Now, we add the result from Step 1 to (3−2i)(3 - 2i):

−3+12i+3−2i=−3+3+12i−2i=0+10i-3 + 12i + 3 - 2i = -3 + 3 + 12i - 2i = 0 + 10i

Conclusion

In this article, we simplified the expression 3×(−1+4i)+(3−2i)3 \times (-1 + 4i) + (3 - 2i) using the properties of complex numbers. We followed the order of operations (PEMDAS) and used the distributive property to multiply 33 by (−1+4i)(-1 + 4i). Finally, we added the result to (3−2i)(3 - 2i) to obtain the simplified expression 0+10i0 + 10i.

Final Answer

The final answer is 10i\boxed{10i}.

Related Topics

  • Complex Numbers: Properties and Operations
  • Simplifying Expressions with Complex Numbers
  • Multiplication and Division of Complex Numbers

References

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Introduction

In our previous article, we simplified the expression 3×(−1+4i)+(3−2i)3 \times (-1 + 4i) + (3 - 2i) using the properties of complex numbers. In this article, we will answer some frequently asked questions related to the simplification of complex expressions.

Q&A

Q: What is the difference between a real number and a complex number?

A: A real number is a number that can be expressed without any imaginary part, whereas a complex number is a number that has both a real part and an imaginary part.

Q: How do you add complex numbers?

A: To add complex numbers, you add the real parts and the imaginary parts separately.

Q: How do you multiply complex numbers?

A: To multiply complex numbers, you use the distributive property and the fact that i2=−1i^2 = -1.

Q: What is the conjugate of a complex number?

A: The conjugate of a complex number is obtained by changing the sign of the imaginary part.

Q: How do you divide complex numbers?

A: To divide complex numbers, you multiply the numerator and denominator by the conjugate of the denominator.

Q: What is the simplified form of the expression 3×(−1+4i)+(3−2i)3 \times (-1 + 4i) + (3 - 2i)?

A: The simplified form of the expression 3×(−1+4i)+(3−2i)3 \times (-1 + 4i) + (3 - 2i) is 0+10i0 + 10i.

Q: Can you provide an example of a complex expression that can be simplified using the properties of complex numbers?

A: Yes, consider the expression 2×(3+4i)+(1−2i)2 \times (3 + 4i) + (1 - 2i). Using the properties of complex numbers, we can simplify this expression as follows:

2×(3+4i)+(1−2i)=6+8i+1−2i=7+6i2 \times (3 + 4i) + (1 - 2i) = 6 + 8i + 1 - 2i = 7 + 6i

Q: How do you simplify an expression with multiple complex numbers?

A: To simplify an expression with multiple complex numbers, you can use the distributive property and the fact that i2=−1i^2 = -1. You can also use the properties of complex numbers, such as the fact that the sum of two complex numbers is equal to the sum of their real parts and imaginary parts.

Conclusion

In this article, we answered some frequently asked questions related to the simplification of complex expressions. We also provided examples of complex expressions that can be simplified using the properties of complex numbers.

Final Answer

The final answer is 10i\boxed{10i}.

Related Topics

  • Complex Numbers: Properties and Operations
  • Simplifying Expressions with Complex Numbers
  • Multiplication and Division of Complex Numbers

References