Problem 116 Why must $a$ and $b$ represe... [FREE SOLUTION]

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College Algebra

Robert Blitzer

$Math Studyset 91影视 Explanations$ Math

3 Edition

Chapter 0: Problem 116

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

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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 by step solution

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.

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.

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.

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.

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91影视

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

Step by step solution

Behavior of square roots for negative numbers

Behavior of the product of square roots

Behavior of cube roots for negative numbers

Behavior of the product of cube roots

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