Problem 127 - Algebra and Trigonometry

Question

Why must \(a\) and \(b\) represent nonnegative 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

For the equation \(\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}\), \(a\) and \(b\) must be nonnegative because the square root of a negative number is not a real number. However, for the equation \(\sqrt[3]{a} \cdot \sqrt[3]{b} = \sqrt[3]{ab}\), \(a\) and \(b\) do not necessarily need to be nonnegative because every real number has a unique real cube root.

Step 1: Understanding square roots of nonnegative numbers

First, consider the square root. By definition, the square root of a number \(x\) (denoted as \(\sqrt{x}\)) is a number whose square (the result of multiplying the number by itself) is \(x\). However, because squaring a real number always yields a nonnegative result, \(\sqrt{x}\) is only defined for nonnegative \(x\). Thus, for \(a\) and \(b\) to fulfill \(\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}\), both \(a\) and \(b\) must be nonnegative numbers. If one or both were negative, the expression would be undefined (as the square root of a negative number is not a real number).

Step 2: Considering the case of cube roots

Moving on to the cube root, by definition, it is a number whose cube (the result of multiplying the number by itself twice) is \(a\). In the real-number domain, every number, positive, negative, or zero, has a unique real cube root. Therefore, \( a \) and \( b \) need not be nonnegative for \(\sqrt[3]{a} \cdot \sqrt[3]{b} = \sqrt[3]{ab}\) to hold true.