Problem 34 - Algebra for College Students

Question

Write each of the following in terms of \(i\) and simplify. For example, $$ \sqrt{-20}=i \sqrt{20}=i \sqrt{4} \sqrt{5}=2 i \sqrt{5} $$ $$\sqrt{-84}$$

Step-by-Step Solution

Answer

\( \sqrt{-84} = 2i \sqrt{21} \)

Step 1: Understand the Problem

We need to express the square root of a negative number in terms of the imaginary unit, which is denoted by \( i \). The imaginary unit \( i \) is defined as \( \sqrt{-1} \). Our task is to simplify \( \sqrt{-84} \) by using \( i \).

Step 2: Express the Negative Square Root

Rewrite the square root expression \( \sqrt{-84} \) in terms of \( i \). You start by recognizing \( \sqrt{-1} \) as \( i \), thus: \[ \sqrt{-84} = \sqrt{-1 \cdot 84} = \sqrt{-1} \cdot \sqrt{84} = i \sqrt{84} \]

Step 3: Simplify the Square Root of 84

Factor 84 into its prime factors to simplify the square root.\[ 84 = 2^2 \times 3 \times 7 \]Use the property of square roots: \( \sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b} \).This allows us to simplify: \[ \sqrt{84} = \sqrt{4 \times 21} = \sqrt{4} \cdot \sqrt{21} = 2\sqrt{21} \]

Step 4: Combine and Simplify Expression

Substitute the simplified square root back into the expression obtained from the earlier step.\[ i \sqrt{84} = i \cdot 2 \sqrt{21} = 2i \sqrt{21} \]So, \( \sqrt{-84} \) simplifies to \( 2i \sqrt{21} \).

Key Concepts

Square Roots

Square roots are fundamental in mathematics. They represent a number which, when multiplied by itself, gives you the original number. For a non-negative number \(n\), \(\sqrt{n}\) is the principal square root. However, if you have a negative number, like \(-84\), this becomes more complex because a real number times itself cannot be negative. This is where imaginary numbers come into play. By defining the imaginary unit as \(i = \sqrt{-1}\), we establish that \(\sqrt{-84}\) offers a way to handle this by separating a negative factor using \(i\). You rewrite \(\sqrt{-84}\) as \(\sqrt{-1 \cdot 84} = i \sqrt{84}\), thus converting the expression into one involving imaginary numbers.

Simplifying Expressions

Simplifying expressions, especially those involving square roots, requires understanding prime factorization and the properties of square roots. When simplifying \(i \sqrt{84}\), you start with the factorization of 84. Breaking it down gives \(84 = 2^2 \times 3 \times 7\). Using the property \(\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}\), you can separate out perfect squares. Here, \(4 = 2^2\) is a perfect square, so \(\sqrt{84} = \sqrt{4 \cdot 21} = 2\sqrt{21}\). This leads us to \(i \sqrt{84} = i \times 2 \sqrt{21} = 2i \sqrt{21}\). Thus, you have simplified \(\sqrt{-84}\) to its most reduced form: \(2i \sqrt{21}\).

Complex Numbers

Complex numbers expand the idea of number systems by incorporating both real and imaginary parts. In this form, a complex number is expressed as \(a + bi\), where \(a\) and \(b\) are real numbers. Imaginary numbers arise when dealing with the square roots of negative numbers, by using the unit \(i\). In our example, by simplifying \(\sqrt{-84}\) to \(2i\sqrt{21}\), we understand it as \(0 + 2i\sqrt{21}\). The real part is zero, and the imaginary part is \(2\sqrt{21}\) multiplied by \(i\). This versatility enables handling a variety of mathematical problems that can't be solved with just real numbers.