Problem 60 Multiply and simplify. Write eac... [FREE SOLUTION]

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Intermediate Algebra : Concepts and Applications

Marvin L. Bittinger, David J. Ellenbogen, Barbara L. Johnson

$Math Studyset 91影视 Explanations$ Math

10 Edition

Chapter 7: Problem 60

Multiply and simplify. Write each answer in the form $a+b i$. $$(1-2 i)^{2}$$

Short Answer

Expert verified

-3 - 4i

Step by step solution

Write the given expression

Write the expression in mathematical form: $(1 - 2i)^2$.

Expand the expression

Use the formula for the square of a binomial $(a - b)^2 = a^2 - 2ab + b^2$. Let $a = 1$ and $b = 2i$. Therefore, $(1 - 2i)^2 = 1^2 - 2 \cdot 1 \cdot 2i + (2i)^2$.

Calculate each term

Calculate the terms:a) $1^2 = 1$b) $-2 \cdot 1 \cdot 2i = -4i$c) $(2i)^2 = 4i^2 = 4(-1) = -4$ because $i^2 = -1$.

Combine the terms

Combine the calculated terms:$1 - 4i - 4$.

Simplify the expression

Simplify the combined terms to get the final form: $1 - 4i - 4 = -3 - 4i$.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Expanding Binomials

When expanding binomials, we use a specific formula for the square of a binomial. For any binomials of the form $a - b$, the square $ (a - b)^2 $ can be expanded using the formula:

$ (a - b)^2 = a^2 - 2ab + b^2 $

To better understand, let's apply it to the actual binomial in the exercise, $1 - 2i $. Identifying $a = 1 $ and $b = 2i $, we substitute these values into the expansion formula:

$(1 - 2i)^2 = 1^2 - 2 \cdot 1 \cdot 2i + (2i)^2$

Imaginary Unit i

The imaginary unit $i$ is crucial in complex number operations. It has a defining property:

$i^2 = -1$

When working with complex numbers, recognizing this property is essential. In the exercise, we encounter $(2i)^2$. Using the property of $i$, this becomes:

$ (2i)^2 = 4i^2 $
Since $i^2 = -1$, it transforms to $4 \cdot (-1) = -4 $

Understanding and applying this property makes simplifying expressions that involve \i\ straightforward.

Simplifying Complex Numbers

Simplifying complex numbers involves combining and reducing all real and imaginary components. From the exercise, after expanding, we have:

$1 - 4i - 4$

We then combine the real parts and the imaginary parts separately. Combining $1$ and $-4$ gives us:

$1 - 4 = -3$

So the expression can be simplified to:

$-3 - 4i$

This final form, $a + bi$, is standard for expressing complex numbers. It ensures we present the complex number in its simplest form.

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91影视

Multiply and simplify. Write each answer in the form \(a+b i\). $$(1-2 i)^{2}$$

Short Answer

Step by step solution

Write the given expression

Expand the expression

Calculate each term

Combine the terms

Simplify the expression

Key Concepts

Expanding Binomials

Imaginary Unit i

Simplifying Complex Numbers

One App. One Place for Learning.

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