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Chapter 8: Problem 52

Find the following products. \((2+7 i)(2-7 i)\)

Short Answer

Expert verified
(2+7i)(2-7i) = 53.

Step by step solution

01

Recognize the Form

The expression \((2+7i)(2-7i)\) is in the form \((a+bi)(a-bi)\), which is known as the difference of squares.
02

Apply the Difference of Squares Identity

The difference of squares states that \((a+bi)(a-bi) = a^2 - (bi)^2\). Here, \(a=2\) and \(b=7\).
03

Calculate \(a^2\)

Calculate \(a^2 = 2^2 = 4\).
04

Calculate \((bi)^2\)

Calculate \((bi)^2 = (7i)^2 = 49i^2\). Recall that \(i^2=-1\), so \(49i^2 = 49(-1) = -49\).
05

Substitute and Simplify

Substitute the values into the difference of squares formula: \(a^2 - (bi)^2 = 4 - (-49)\). Simplify to obtain \(4 + 49 = 53\).

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Key Concepts

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

Difference of Squares
The difference of squares is a powerful algebraic identity that can make multiplying certain expressions much easier. When we see an expression like \((a+bi)(a-bi)\), it fits perfectly into this identity.
  • It is called the "difference of squares" because it involves two terms squared and their difference: \(a^2 - (bi)^2\).
  • This formula works for any two numbers or expressions of this form, not just real numbers.
  • It simplifies the multiplication by turning it into a subtraction problem, which is often easier to compute.
In this exercise, we have \((2+7i)(2-7i)\), which becomes \(2^2 - (7i)^2\). This simplification makes it easy to calculate as it reduces the complexity of multiplication.
Imaginary Unit
The imaginary unit, denoted as \(i\), is a fundamental concept in complex numbers. It is defined by the property \(i^2 = -1\).
  • This means that whenever we see \(i^2\) in an expression, we can replace it with \(-1\).
  • This substitution is key to simplifying complex number expressions, especially when they involve powers of \(i\).
  • In the difference of squares problem, we encounter \((7i)^2\). By applying the property of the imaginary unit, this becomes \(49i^2 = 49(-1) = -49\).
Understanding how the imaginary unit works allows us to manipulate expressions involving complex numbers effectively and efficiently.
Complex Multiplication
Complex multiplication involves multiplying two complex numbers in the standard form \(a+bi\). This process combines both the real and imaginary parts each number has.
  • Each term in the first complex number multiplies by each term in the second complex number, resulting in four products.
  • These products are then combined, and any powers of \(i\) are simplified using the property \(i^2 = -1\).
Using the difference of squares identity and properties of the imaginary unit, we can simplify this multiplication significantly. Instead of considering four separate products, the identity \((a+bi)(a-bi) = a^2 - (bi)^2\) reduces it to two simpler calculations: \(a^2\) and \((bi)^2\). Hence, \((2+7i)(2-7i)\) simplifies to \(4 - (-49) = 53\). By converting the operation to a subtraction of squares, we make the process quicker and easier.

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Most popular questions from this chapter

Show that \(x=2+3 i\) is a solution to the equation \(x^{2}-4 x+13=0\).

Use a calculator to help write each complex number in standard form. Round the numbers in your answers to the nearest hundredth.\(1 \operatorname{cis} 205^{\circ}\)

Graph each complex number. In each case, give the absolute value of the number.\(1+i\)

Graph each complex number along with its opposite and conjugate.\(4 i\)

Graph each complex number along with its opposite and conjugate.\(-5-2 i\)
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