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Chapter 1: Problem 19

Find each product and write the result in standard form. $$ (2+3 i)^{2} $$

Short Answer

Expert verified
The result in standard form is \(-5 + 12i\).

Step by step solution

01

Multiplication of Complex Numbers

We begin by squaring \((2+3i)\) through multiplying the complex number by itself: \((2+3i) (2+3i)\). We then apply the distributive law (also known as expansion), to get \((2*2 + 2*3i + 3i*2 + (3i)^2)\).
02

Apply rules for real and imaginary part

Then we transform the obtained expression into: \(4 + 6i + 6i + 9i^2\). Don't forget that \(i=\sqrt{-1}\) and \(i^{2}=-1\).
03

Simplify the result

Finally, get the result in standard form, by performing the multiplications and combining like terms: \(4 + 12i - 9\). This simplifies to \(-5 + 12i\)

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Key Concepts

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

Standard Form
Complex numbers are written in standard form as \(a + bi\), where \(a\) is the real part and \(bi\) is the imaginary part. The imaginary unit \(i\) represents \(\sqrt{-1}\). Writing complex numbers in standard form is essential for ease of reading and further calculations.
  • Identify the real part, which is always a number without the imaginary unit \(i\).
  • Identify the imaginary part, which includes the imaginary unit \(i\).

After operations with complex numbers, results need to be arranged in this format. For example, after simplifying \(4 + 12i - 9\), we arrange it as \(-5 + 12i\), where \(-5\) is the real part, and \(12i\) is the imaginary part.
i Squared Rule
In complex numbers, the imaginary unit \(i\) has a special property: \(i^2 = -1\). This property is fundamental in simplifying expressions that involve complex numbers. Understanding this rule is key to dealing with powers of \(i\).
  • Recognize that \(i^2 = -1\).
  • This property helps transform imaginary terms into real terms during simplification.
  • Use it whenever you encounter \(i^2\) in an expression.

For example, in the expression \(9i^2\), by applying \(i^2 = -1\), it becomes \(9(-1) = -9\). This step is crucial in ensuring all parts are correctly simplified in the process.
Distributive Law
The distributive law is a basic algebraic principle that allows you to expand expressions. It says that for any numbers or expressions \(a\), \(b\), and \(c\), the expression \(a(b + c)\) is equal to \(ab + ac\). In complex numbers, it works the same way.
  • Apply it when multiplying two binomial expressions.
  • Multiply each term in the first expression with every term in the second expression.
  • Combine like terms to simplify the expression.

For example, when squaring \((2+3i)\), use this law to multiply each part: \((2+3i)(2+3i) = 2\times2 + 2\times3i + 3i\times2 + (3i)^2\). Each part is multiplied individually, ensuring no term is missed in simplification.

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Most popular questions from this chapter

Solve absolute value inequality. \(2>|1-x|\)

Use interval notation to express solution sets and graph each solution set on a number line. Solve linear inequality. \(-4(x+2)>3 x+20\)

Solve absolute value inequality. \(|x-1| \geq 2\)

Solve absolute value inequality. \(5>|4-x|\)

When 3 times a number is subtracted from \(4,\) the absolute value of the difference is at least \(5 .\) Use interval notation to express the set of all numbers that satisfy this condition.
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