Problem 116 - College Algebra

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

a and b must be nonnegative for \(\sqrt{a} \cdot \sqrt{b}=\sqrt{ab}\) because square roots of negative numbers are not defined in the real number system. However, in the case of \(\sqrt[3]{a} \cdot \sqrt[3]{b}=\sqrt[3]{ab}\), this restriction is not necessary because cube roots are defined for negative numbers.

Step 1: Behavior of square roots for negative numbers

As a step of this solution, first consider the behavior of the square root function. The square root of a number can't be negative because by definition, a square root of a number x is a value that, when multiplied by itself, gives x. There's no real number that can be squared (multiplied by itself) to yield a negative number - this is because both a positive number squared and a negative number squared will yield a positive result. Hence, square roots of negative numbers are not defined in the real numbers. With this understanding, the expression \(\sqrt{a} \cdot \sqrt{b}\) would be undefined if either a or b were negative.

Step 2: Behavior of the product of square roots

Now consider the product \(\sqrt{a} \cdot \sqrt{b}=\sqrt{ab}\). As per the laws of multiplication of square roots, the square root of the product of two numbers is equal to the product of their square roots, provided that a and b are nonnegative. If either a or b is negative, \(\sqrt{a} \cdot \sqrt{b}\) would be undefined, but \(\sqrt{ab}\) would still have a valid value. Therefore, this equation is only valid when a and b are both nonnegative.

Step 3: Behavior of cube roots for negative numbers

Moving on to the second part of the exercise, consider the behavior of the cube root function. The cube root of a negative number is defined, unlike the square root. This is because a negative number can be cubed (multiplied by itself twice) to yield a negative number (because the product of three negative numbers is negative). Therefore, the cube root function can accept negative numbers.

Step 4: Behavior of the product of cube roots

Looking at the second expression, we have \(\sqrt[3]{a} \cdot \sqrt[3]{b}=\sqrt[3]{ab}\). Here, even if either a or b is negative, both sides of the equation are well-defined (because, as explained before, cube roots of negative numbers are allowed). Therefore, for this equation, a and b do not need to be nonnegative.