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Chapter 0: Problem 127

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.

Short Answer

Expert verified
The restriction that \(a\) and \(b\) must be nonnegative is necessary for the equation \( \sqrt{a} \cdot \sqrt{b} = \sqrt{ab} \) to be true because you can't have square roots of negative numbers in the set of real numbers. However, this restriction is not needed for the equation \( \sqrt[3]{a} \cdot \sqrt[3]{b} = \sqrt[3]{ab} \) because negative numbers do have real cube roots.

Step by step solution

01

Understanding Square Roots

A square root of a number is a value that, when multiplied by itself, gives the original number. When dealing with real numbers, negatives do not have a real square root because multiplying two negatives together results in a positive.
02

Explaining \( \sqrt{a} \cdot \sqrt{b} = \sqrt{ab} \)

When \(a\) and \(b\) are nonnegative, the expression \( \sqrt{a} \cdot \sqrt{b} = \sqrt{ab} \) holds true because the product of the square roots is equal to the square root of the product. However, if either \(a\) or \(b\) is negative, the left side of the expression is undefined in the real number system, as you can't take the square root of a negative number.
03

Understanding Cube Roots

A cube root of a number is a value that, when multiplied by itself twice (i.e. cubed), gives the original number. The cube root of a negative number is negative, which differentiates it from a square root.
04

Explaining \( \sqrt[3]{a} \cdot \sqrt[3]{b} = \sqrt[3]{ab} \)

In the case of \( \sqrt[3]{a} \cdot \sqrt[3]{b} \), there is no need for the restriction that \(a\) and \(b\) be nonnegative. Because negative numbers do have real cube roots, this expression holds true whether \(a\) and \(b\) are negative, positive, or zero.

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