Problem 37 - Algebra for College Students

Question

Write each of the following in terms of \(i\) and simplify. For example, $$ \sqrt{-20}=i \sqrt{20}=i \sqrt{4} \sqrt{5}=2 i \sqrt{5} $$ $$3 \sqrt{-28}$$

Step-by-Step Solution

Answer

The simplified expression is \(6i\sqrt{7}\).

Step 1: Identify the Negative Inside the Square Root

The expression given is \(3 \sqrt{-28}\). To express the square root of a negative number, we identify the negative sign and note that \( \sqrt{-1} = i \). Therefore, \( \sqrt{-28} = \sqrt{28} \cdot \sqrt{-1} = i \sqrt{28} \).

Step 2: Factor the Number Inside the Square Root

We factor \( 28 \) into its prime components: \( 28 = 4 \times 7 \). So \( \sqrt{28} = \sqrt{4 \cdot 7} \).

Step 3: Simplify the Square Root

Use the property of square roots: \( \sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b} \). Thus, \( \sqrt{4 \cdot 7} = \sqrt{4} \cdot \sqrt{7} = 2 \sqrt{7} \), since \( \sqrt{4} = 2 \).

Step 4: Combine and Simplify with the Multiplier

Now substitute back into the original expression: \( 3 \sqrt{-28} = 3 \cdot i \cdot 2 \sqrt{7} \). Calculate the product: \( 3 \cdot 2 \cdot i \cdot \sqrt{7} = 6i \sqrt{7} \). This is the simplified expression.

Key Concepts

Imaginary Unit

When dealing with the square root of a negative number, you encounter the concept of the imaginary unit, denoted as \( i \). This is a fundamental building block in complex numbers, used to make sense of these challenging math situations.

• The imaginary unit \( i \) is defined by the equation \( i^2 = -1 \). This means that \( i \) is the square root of \(-1\).
• Using \( i \) allows us to represent any square root of a negative number as \( i \) times the square root of the corresponding positive number.
• For example, \( \sqrt{-28} = i \sqrt{28} \).

Understanding \( i \) provides a way to extend our number system to include solutions to equations that do not have real solutions. It's a critical concept in the study of complex numbers, allowing us to work with and perform calculations on negative square roots.

Simplifying Square Roots

Simplifying square roots is an essential skill in mathematics that involves breaking down a root to its simplest form. To do this effectively, you need to recognize and factor the number inside the root.

• The process starts with factoring the number inside the square root into its prime factors.
• If you can form any perfect squares from these factors, you can simplify the square root further.
• For example, with the number 28 inside the root, you can factor it as \( 28 = 4 \times 7 \) and then simplify \( \sqrt{28} = \sqrt{4 \times 7} = \sqrt{4} \cdot \sqrt{7} = 2\sqrt{7} \).

By mastering this simplification process, you're able to manage and simplify complex expressions, making it possible to express solutions in the simplest and most useful form.

Prime Factorization

Prime factorization is the process of breaking down a number into its smallest divisible components, known as prime numbers. This technique is incredibly useful for simplifying square roots and understanding the structure of numbers.

• A prime number is a natural number greater than 1 that cannot be divided by any number other than 1 and itself. Examples include 2, 3, 5, and 7.
• To factor a number like 28, you divide it by the smallest possible prime numbers. For instance, \( 28 \div 2 = 14 \), and \( 14 \div 2 = 7 \), leaving us with the factors \( 2, 2, \) and \( 7 \).
• This leads to the expression \( 28 = 2^2 \times 7 \).

Prime factorization turns complex expressions into easily manageable parts and is a cornerstone of simplifying square roots. Once you have the prime factors, you can easily identify perfect squares and simplify square roots efficiently.