Simplify The Expression To { A + Bi$}$ Form:${(11 - 5i)(-1 + 2i)}$

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Introduction

In mathematics, simplifying complex expressions is a crucial skill that helps us solve problems in algebra, geometry, and other branches of mathematics. One of the most common forms of complex numbers is the {a + bi$}$ form, where {a$}$ is the real part and {bi$}$ is the imaginary part. In this article, we will simplify the expression (11−5i)(−1+2i){(11 - 5i)(-1 + 2i)} to the {a + bi$}$ form.

Understanding Complex Numbers

Before we dive into simplifying the expression, let's quickly review what complex numbers are. A complex number is a number that can be expressed in the form {a + bi$}$, where {a$}$ is the real part and {bi$}$ is the imaginary part. The real part {a$}$ is a real number, and the imaginary part {bi$}$ is a multiple of the imaginary unit {i$}$, where {i$}$ is defined as the square root of -1.

Distributive Property

To simplify the expression (11−5i)(−1+2i){(11 - 5i)(-1 + 2i)}, we will use the distributive property, which states that for any complex numbers {a + bi$}$ and {c + di$}$, the product (a+bi)(c+di){(a + bi)(c + di)} can be expanded as:

(a+bi)(c+di)=ac+adi+bci+bdi2{(a + bi)(c + di) = ac + adi + bci + bdi^2}

Since {i^2 = -1$}$, we can simplify the expression further:

(a+bi)(c+di)=ac+adi+bci−bd{(a + bi)(c + di) = ac + adi + bci - bd}

Simplifying the Expression

Now that we have reviewed the distributive property, let's apply it to simplify the expression (11−5i)(−1+2i){(11 - 5i)(-1 + 2i)}. We will multiply each term in the first expression by each term in the second expression:

(11−5i)(−1+2i)=11(−1)+11(2i)−5i(−1)−5i(2i){(11 - 5i)(-1 + 2i) = 11(-1) + 11(2i) - 5i(-1) - 5i(2i)}

Expanding the Terms

Now, let's expand each term:

(11−5i)(−1+2i)=−11+22i+5i−10i2{(11 - 5i)(-1 + 2i) = -11 + 22i + 5i - 10i^2}

Simplifying the Terms

Since {i^2 = -1$}$, we can simplify the expression further:

(11−5i)(−1+2i)=−11+22i+5i+10{(11 - 5i)(-1 + 2i) = -11 + 22i + 5i + 10}

Combining Like Terms

Now, let's combine like terms:

(11−5i)(−1+2i)=(−11+10)+(22i+5i){(11 - 5i)(-1 + 2i) = (-11 + 10) + (22i + 5i)}

Simplifying the Expression

Finally, let's simplify the expression:

(11−5i)(−1+2i)=−1+27i{(11 - 5i)(-1 + 2i) = -1 + 27i}

Conclusion

In this article, we simplified the expression (11−5i)(−1+2i){(11 - 5i)(-1 + 2i)} to the {a + bi$}$ form. We used the distributive property to expand the expression and then combined like terms to simplify it. The final simplified expression is −1+27i{-1 + 27i}.

Frequently Asked Questions

  • What is the distributive property? The distributive property is a mathematical property that allows us to expand the product of two complex numbers.
  • How do I simplify a complex expression? To simplify a complex expression, you can use the distributive property to expand the expression and then combine like terms.
  • What is the imaginary unit? The imaginary unit is a mathematical constant denoted by {i$}$ and defined as the square root of -1.

Final Answer

The final answer is: −1+27i\boxed{-1 + 27i}

Introduction

Complex numbers are a fundamental concept in mathematics, and understanding them is crucial for solving problems in algebra, geometry, and other branches of mathematics. In our previous article, we simplified the expression (11−5i)(−1+2i){(11 - 5i)(-1 + 2i)} to the {a + bi$}$ form. In this article, we will answer some frequently asked questions about complex numbers and provide additional information to help you better understand this topic.

Q&A: Complex Numbers

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 can be expressed in the form {a + bi$}$, where {a$}$ is the real part and {bi$}$ is the imaginary part.

Q: What is the imaginary unit?

A: The imaginary unit is a mathematical constant denoted by {i$}$ and defined as the square root of -1. It is used to represent the imaginary part of a complex number.

Q: How do I simplify a complex expression?

A: To simplify a complex expression, you can use the distributive property to expand the expression and then combine like terms. You can also use the fact that {i^2 = -1$}$ to simplify the expression further.

Q: What is the conjugate of a complex number?

A: The conjugate of a complex number {a + bi$}$ is {a - bi$}$. The conjugate of a complex number is used to simplify expressions and to find the absolute value of a complex number.

Q: How do I find the absolute value of a complex number?

A: To find the absolute value of a complex number {a + bi$}$, you can use the formula ∣a+bi∣=a2+b2{|a + bi| = \sqrt{a^2 + b^2}}. The absolute value of a complex number is a measure of its distance from the origin in the complex plane.

Q: What is the polar form of a complex number?

A: The polar form of a complex number {a + bi$}$ is r(cos⁡θ+isin⁡θ){r(\cos \theta + i \sin \theta)}, where {r$}$ is the absolute value of the complex number and {\theta$}$ is the angle between the complex number and the positive real axis.

Q: How do I convert a complex number from rectangular form to polar form?

A: To convert a complex number from rectangular form to polar form, you can use the formulas r=a2+b2{r = \sqrt{a^2 + b^2}} and θ=tan⁡−1(ba){\theta = \tan^{-1} \left(\frac{b}{a}\right)}. You can also use the fact that cos⁡θ=ar{\cos \theta = \frac{a}{r}} and sin⁡θ=br{\sin \theta = \frac{b}{r}} to find the polar form of the complex number.

Conclusion

In this article, we answered some frequently asked questions about complex numbers and provided additional information to help you better understand this topic. We hope that this article has been helpful in clarifying any doubts you may have had about complex numbers.

Final Answer

The final answer is: Thereisnofinalnumericalanswertothisarticle.\boxed{There is no final numerical answer to this article.}