Problem 34 - Intermediate Algebra

Question

Write each of the following in terms of \(i\) and simplify. $$ \sqrt{-84} $$

Step-by-Step Solution

Answer

\( 2i \sqrt{21} \)

Step 1: Identify Imaginary Component

Identify that the square root of a negative number involves imaginary numbers. \( \sqrt{-84} \) can be rewritten as \( \sqrt{84} \times \sqrt{-1} \). This is the definition of imaginary numbers, where \( \sqrt{-1} = i \).

Step 2: Simplify the Square Root of the Negative Component

Express the square root involving the imaginary unit: \( \sqrt{-84} = \sqrt{84} \cdot i \).

Step 3: Simplify the Square Root of the Positive Component

Find \( \sqrt{84} \) by breaking it down into its prime factors. The prime factors of 84 are 2, 2, 3, and 7. This can be expressed as \( \sqrt{2^2 \times 3 \times 7} \).

Step 4: Simplify Expression Further Using Prime Factorization

Since \( 2^2 = 4 \) is a perfect square, we can simplify \( \sqrt{4} = 2 \). Therefore, \( \sqrt{84} \) becomes \( \sqrt{2^2 \times 3 \times 7} = 2 \times \sqrt{21} \).

Step 5: Combine and Simplify the Expression with the Imaginary Unit

Combine the simplifications to express the square root in terms of \( i \). \( \sqrt{-84} = 2 \cdot \sqrt{21} \cdot i \). Simplifying it, we get \( 2i \sqrt{21} \).

Key Concepts

Complex Numbers

Complex numbers are an essential mathematical concept that extends the idea of the real number line. These numbers are expressed in the form of \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit. Understanding complex numbers is like opening a new dimension beyond the regular number line that we are used to.

• Real Part: In the expression \(a + bi\), the real part is \(a\). It resides on the horizontal axis of the complex plane.
• Imaginary Part: The \(bi\) component is the imaginary part. The \(i\) symbolizes the square root of -1, and it is plotted on the vertical axis.

By combining these parts, complex numbers allow us to solve equations that have no real number solutions, such as \(x^2 + 1 = 0\). In such cases, the answer involves imaginary numbers, where the root is \(x = i\). This interplay between the real and imaginary components is what makes complex numbers so powerful and useful in various fields of science and engineering.

Square Roots

Square roots can sometimes be tricky, especially when dealing with negative numbers. Typically, the square root of a number \(x\) is the value \(y\) such that \(y^2 = x\). However, when \(x\) is negative, we cannot find a real number \(y\) that satisfies this property.The introduction of imaginary numbers resolves this issue. Each negative number can be expressed as \(-1\) times its positive counterpart, e.g. \(-84 = 84 \times -1\). Thus, the square root of \(-84\) is split into two components:

• \(\sqrt{-84} = \sqrt{84} \times \sqrt{-1}\)
• \(\sqrt{-1} = i\), the imaginary unit

Therefore, the square root of \(-84\) becomes \(i\sqrt{84}\). This approach allows us to work with square roots of negative numbers more flexibly and connect them to complex roots.

Prime Factorization

Prime factorization is a method used to break down a number into a product of its prime factors. This technique comes in handy when simplifying square roots of numbers, such as \(\sqrt{84}\).By breaking 84 down into its prime components, we find:

• 84 can be expressed as \(2 \times 2 \times 3 \times 7\)
• This is \(2^2 \times 3 \times 7\), showing the prime factors involved

Knowing which components are perfect squares is helpful. Here, \(2^2\) is a perfect square, which means \(\sqrt{4} = 2\). This gives us a more simplified form: \(\sqrt{2^2 \times 3 \times 7} = 2\sqrt{21}\). By using prime factorization, numbers under the square root can be simplified with ease. This simplification is especially useful when dealing with expressions involving imaginary numbers, like \(\sqrt{-84}\), making it more manageable as \(2i\sqrt{21}\).