Problem 17 - A Graphical Approach to College Algebra

Question

Answer each question. For which of the following cases is \(\sqrt{a b}=\sqrt{a} \cdot \sqrt{b}\) a true statement? A. \(a\) and \(b\) both positive B. \(a\) and \(b\) both negative

Step-by-Step Solution

Answer

The equation is true for Case A: both \(a\) and \(b\) positive.

Step 1: Understand square root properties

The equation \(\sqrt{a b} = \sqrt{a} \cdot \sqrt{b}\) is generally true when both \(a\) and \(b\) are non-negative. This is based on the property of square roots that allows the splitting of the square root of a product into the product of square roots.

Step 2: Analyze Case A

In Case A, both \(a\) and \(b\) are positive. Since both numbers are non-negative, the equation \(\sqrt{a b} = \sqrt{a} \cdot \sqrt{b}\) holds true by the properties of square roots.

Step 3: Analyze Case B

In Case B, both \(a\) and \(b\) are negative. The square root of a negative number is not defined in the real number system. Therefore, \(\sqrt{a} \) or \( \sqrt{b} \) do not exist in real numbers, making this scenario impossible to satisfy the equation in real numbers.

Key Concepts

Properties of Square Roots

Square roots have a unique set of properties that define how they interact with numbers. One of the fundamental properties is that the square root of a product can be decomposed into the product of individual square roots. Mathematically, this is expressed as:

• \(\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}\)

This property holds true under specific conditions, namely when both \(a\) and \(b\) are non-negative numbers.
- **Positive Numbers**: When both numbers are positive, you can safely apply this property. For instance, \(\sqrt{3 \cdot 4} = \sqrt{3} \cdot \sqrt{4}\) holds true. - **Negative Numbers**: If either \(a\) or \(b\) is negative, or both, the property fails because the square root of a negative number isn't addressed in the realm of real numbers.
Understanding this allows you to simplify and solve square root expressions more effectively.

Real Numbers

The real number system consists of a wide array of numbers including integers, fractions, and decimals, that can be positive, negative, or zero. Real numbers are what we use in everyday mathematics and when dealing with square roots in this domain, it is important to remember:

• Square roots of non-negative numbers always result in real numbers.
• Negative numbers, within the scope of real numbers, do not have real square roots, as their square roots fall into the imaginary number category.

For example, \(\sqrt{16} = 4\), which is a real number. However, \(\sqrt{-16}\) does not exist in the real number line, as there is no real number that squared gives \(-16\).
This fundamental distinction greatly influences how we calculate and understand square roots within real numbers.

Negative Numbers

Negative numbers are a vital part of the number system representing values less than zero. While you can perform many arithmetic operations such as addition and subtraction with them, square roots introduce a unique challenge.

• The square root of a negative number isn't defined within the real numbers.
• Such numbers require the use of imaginary numbers to describe their square roots.

For example, the square root of \(-9\) is expressed as \(3i\) in the imaginary number space, where \(i\) is the imaginary unit equal to \(\sqrt{-1}\).
Understanding this concept is fundamental when exploring more advanced mathematics or when dealing with equations where negative roots might arise. It explains why equations like \(\sqrt{a \cdot b} = \sqrt{a}\cdot\sqrt{b}\) do not hold true when involving negative numbers, making imaginary numbers a necessary consideration.