Problem 127 - College Algebra

Question

Why must \(a\) and \(b\) represent non negative numbers when we write \(\sqrt{a} \cdot \sqrt{b}=\sqrt{a b ?}\) Is it necessary to use this restriction in the case of \(\sqrt[3]{a} \cdot \sqrt[3]{b}=\sqrt[3]{a b} ?\) Explain.

Step-by-Step Solution

Answer

The numbers \(a\) and \(b\) have to be non-negative when we write \(\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}\) to ensure that the square root of a negative number, which does not exist in the real number system, is not taken. However, this restriction is not necessary in the case of the cube root operation, \(\sqrt[3]{a} \cdot \sqrt[3]{b} = \sqrt[3]{ab}\), because both positive and negative numbers have real cube roots.

Step 1: Understanding Square Roots

A fundamental property of square roots is that \(\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}\) given that \(a\) and \(b\) are non-negative. The reason for this restriction is because a square root operation implies that the radicand (the number under the root) has to be a positive number or zero. The reason is that in the real number system, if you square any real number, either positive or negative, you always get a positive result or zero. Thus, the square root of a negative number is not a real number, hence the need for the restriction.

Step 2: Understanding Cube Roots

A cube root operation has no such restriction as the square root. We say that \(\sqrt[3]{a} \cdot \sqrt[3]{b} = \sqrt[3]{ab}\), regardless of the value of \(a\) and \(b\). This is because a real number cubed can produce both negative and positive results. Therefore, both positive and negative numbers have real cube roots.

Key Concepts

Non-Negative Numbers

Square roots are something most students encounter early in their math journey. Whenever you see the symbol \(\sqrt{}\), it suggests finding a special number which, when multiplied by itself, gives you the original value you are working with. But here’s the trick: square roots do not cover negative numbers.
A square root only applies to non-negative numbers, meaning zero and positive numbers, because you can't have a real square root if the number inside is negative.

• Zero: \(\sqrt{0} = 0\)
• Positive numbers: \(\sqrt{4} = 2\) because \(2 \times 2 = 4\).

No negative number can have a real square root because no number multiplied by itself equals a negative. You’ll end up with something outside of the real number system, called an imaginary number.

That’s why the condition of being non-negative is crucial in expressions like \(\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}\), ensuring everything stays within the real numbers.

Cube Roots

Cube roots open up possibilities that square roots don't. When you see \(\sqrt[3]{}\), it's about finding a number that multiplies by itself twice to give you the original value. What's interesting with cube roots is there’s no restriction on positivity.
This means both positive and negative numbers qualify for cube roots.

• Positive example: \(\sqrt[3]{8} = 2\) since \(2 \times 2 \times 2 = 8\).
• Negative example: \(\sqrt[3]{-8} = -2\) because \((-2) \times (-2) \times (-2) = -8\).

Negative or positive, it doesn’t matter, cube roots allow more flexibility. So when you multiply the cube roots, like \(\sqrt[3]{a} \cdot \sqrt[3]{b} = \sqrt[3]{ab}\), there are no restrictions.

This inclusive nature of cube roots fits well in the real number system, making calculations easier and more comprehensive.

Real Number System

The real number system is like the home for most numbers you'll deal with in typical math scenarios. It includes every number that can sit on the number line: natural numbers, whole numbers, integers, rational numbers, and irrational numbers. This system excludes imaginary numbers.
Square roots refer to real numbers and work with non-negative values to maintain real results.

The real number system’s rules are simple:

• With square roots, negative results don’t sit on the number line, which is why we need non-negative numbers for them to exist in reality.
• With cube roots, though, negative results align with the system's ability to cover all such values.

This welcoming nature of the real number system provides the logic we see in square and cube root operations.

Simply put, when navigating through square and cube roots, understanding where these numbers fall in the real number system helps clarify their restrictions and allowances.