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Chapter 8: Problem 9

Write in terms of \(i\) and then simplify. \(\sqrt{-4} \cdot \sqrt{-9}\)

Short Answer

Expert verified
The simplified result is \(-6\).

Step by step solution

01

Express in terms of imaginary numbers

The square root of a negative number can be expressed in terms of the imaginary unit, denoted by \(i\), where \(i^2 = -1\). So, \(\sqrt{-4} = \sqrt{4} \cdot \sqrt{-1} = 2i\) and \(\sqrt{-9} = \sqrt{9} \cdot \sqrt{-1} = 3i\).
02

Multiply the terms

Now multiply the expressions obtained: \(2i \cdot 3i\). This gives \(2 \cdot 3 \cdot i^2 = 6i^2\).
03

Simplify the expression using properties of i

Recall that \(i^2 = -1\). Thus, substituting \(-1\) for \(i^2\) in the expression \(6i^2\) gives \(6(-1) = -6\).

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Key Concepts

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

Imaginary Unit
The imaginary unit is a fundamental concept when dealing with square roots of negative numbers. It is denoted by the symbol \( i \), which is defined by the equality \( i^2 = -1 \). This means that \( i \) behaves differently from ordinary numbers. For example:
  • \( i \) is not a real number; it is used to form complex numbers.
  • Square roots of negative numbers become feasible with the help of \( i \).
When we encounter a negative inside a square root, we can express it using the imaginary unit. For instance, \( \sqrt{-1} = i \). This allows us to navigate through calculations such as \( \sqrt{-4} \) and \( \sqrt{-9} \) by breaking them into smaller, manageable components.
Simplifying Expressions
Simplifying expressions that involve imaginary numbers requires a step-by-step approach. When you simplify, you're essentially looking to streamline the expression into a clearer form.
  • Start with recognizing the imaginary numbers. For example, convert \( \sqrt{-4} \) into \( 2i \) and \( \sqrt{-9} \) into \( 3i \).
  • Once converted, these can be multiplied as ordinary terms, keeping in mind that \( i^2 = -1 \).
Multiplication of these terms can seem tricky, but remember:
  • Multiply coefficients: \( 2 \times 3 = 6 \).
  • Apply \( i \times i = i^2 \), and then use the property \( i^2 = -1 \).
By following these steps, you can simplify expressions and achieve an understandable result. The expression \( 2i \cdot 3i \) simplifies to \( 6i^2 \), which breaks down further to \(-6\).
Square Roots of Negative Numbers
Dealing with square roots of negative numbers is a common challenge in algebra. Normally, you cannot take the square root of a negative number in the realm of real numbers. However, complex numbers allow us to expand our understanding.
  • Negative numbers beneath a square root can be rewritten using \( i \), for instance, \( \sqrt{-k} = \sqrt{k} \cdot i \), where \( k \) is a positive real number.
  • This transformation is crucial for operations beyond basic arithmetic.
In practical terms:
  • For \( \sqrt{-4} \), break it down to \( \sqrt{4} \times \sqrt{-1} \), leading to \( 2i \).
  • Similarly, \( \sqrt{-9} \) becomes \( 3i \).
By leveraging the power of the imaginary unit \( i \), square roots of negative numbers become not just solvable, but a routine part of algebraic manipulation. This expands our toolbox for solving equations and understanding complex expressions.

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