Problem 52 - Algebra for College Students

Question

Write each of the following in terms of \(i\), perform the indicated operations, and simplify. For example, $$ \begin{aligned} \sqrt{-3} \sqrt{-8} &=(i \sqrt{3})(i \sqrt{8}) \\ &=i^{2} \sqrt{24} \\ &=(-1) \sqrt{4} \sqrt{6} \\ &=-2 \sqrt{6} \end{aligned} $$ $$\sqrt{-3} \sqrt{-15}$$

Step-by-Step Solution

Answer

\(-3 \sqrt{5}\)

Step 1: Express Each Square Root in Terms of i

To express the square roots in terms of \(i\), recall that \(i = \sqrt{-1}\). Therefore, \(\sqrt{-3} = i \sqrt{3}\) and \(\sqrt{-15} = i \sqrt{15}\).

Step 2: Multiply the Expressions

Multiply the expressions obtained from Step 1:\[(i \sqrt{3})(i \sqrt{15}) = i^2 \sqrt{3} \sqrt{15}\]

Step 3: Simplify Using i^2 = -1

Since \(i^2 = -1\), substitute this into the expression: \[i^2 \sqrt{3} \sqrt{15} = (-1) \sqrt{3 \times 15}\]

Step 4: Simplify the Square Root

Compute the multiplication inside the square root:\[\sqrt{3 \times 15} = \sqrt{45}\]Further, simplify \(\sqrt{45}\):\[\sqrt{45} = \sqrt{9 \times 5} = \sqrt{9} \sqrt{5} = 3\sqrt{5}\]

Step 5: Final Simplification

Substitute the simplified square root back into the expression:\[(-1) \times 3 \sqrt{5} = -3 \sqrt{5}\]

Key Concepts

Square Roots

Square roots are a fundamental concept in mathematics, representing a value that when multiplied by itself gives the original number. For example, the square root of 9 is 3, because 3 multiplied by 3 equals 9. However, when dealing with negative numbers, we enter the realm of complex numbers. Since no real number squared can equal a negative number, this is where imaginary numbers come in. Here's how to handle square roots of negative numbers:

• The square root of a negative number involves imagining the number as a product involving the imaginary unit, represented by the symbol \(i\).
• For example, \(\sqrt{-3}\) can be expressed as \(i \sqrt{3}\), because \(i^2 = -1\).
• When simplifying expressions, convert negative square roots to products of \(i\) and the square root of the positive version of the number.

Understanding how to rewrite these expressions correctly is crucial when performing operations with complex numbers.

Imaginary Unit

The imaginary unit, denoted as \(i\), is a mathematical concept created to solve equations that do not have solutions in the realm of real numbers. It is specifically defined by the property:

• \(i = \sqrt{-1}\), meaning \(i\) squared equals -1, or \(i^2 = -1\).

This simple yet profound definition allows us to extend our number system to include complex numbers, defined as numbers of the form \(a + bi\) where \(a\) and \(b\) are real numbers.

• When working with complex numbers, treat \(i\) as an algebraic variable until you simplify the expression by replacing \(i^2\) with -1.
• Any equation that results in negative square roots will make use of the imaginary unit \(i\) for its simplification.

This ability to incorporate \(i\) helps in performing computations and solving equations that wouldn't be manageable with just real numbers.

Simplification of Expressions

Simplification of expressions in mathematics, especially involving complex numbers, is the process of transforming these expressions into a more manageable or interpretable form. Here are steps to simplify expressions involving complex numbers and \(i\):

• Replace any square roots of negative numbers with expressions involving \(i\). For instance, \(\sqrt{-3} = i\sqrt{3}\).
• When multiplying complex numbers, distribute \(i\) and calculate \(i^2\) as -1 whenever it appears in the product.
• Perform multiplication inside any square root first for further simplification. For instance, simplify \(\sqrt{3 \times 15}\) to \(\sqrt{45}\).
• Break down the product under the square root into its simplest integer factors, such as \(\sqrt{45} = \sqrt{9} \cdot \sqrt{5} = 3\sqrt{5}\).
• Finally, apply the minus sign from \(-1\) as needed to complete the simplification.

These simplification rules are key to reducing complex expressions to their simplest forms, ensuring easier manipulation and understanding.

Mathematical Operations

Performing mathematical operations with complex numbers, particularly those involving the imaginary unit \(i\), requires careful attention to arithmetic rules. Here's a breakdown:

• When handling expressions involving \(i\), always start by expressing everything in terms of \(i\) if they aren't already.
• In problems involving products like \((i \sqrt{3})(i \sqrt{15})\), follow these steps:
  • Multiply the \(i\) terms separately, applying \(i^2 = -1\) as necessary.
  • Multiply the square root terms, simplify as needed.
• Simplify using properties of roots, such as \(\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}\).

These steps ensure you can perform operations correctly and simplify the result to the most digestible form. Knowing these operations helps in seamlessly working through problems involving real and imaginary components.