Problem 109 - Intermediate Algebra

Question

Answer true or false. Assume all radicals represent nonzero real numbers. $$ \sqrt[n]{a} \cdot \sqrt[n]{b}=\sqrt[n]{a b} $$

Step-by-Step Solution

Answer

True.

Step 1: Identify the Rule

The equation given is \( \sqrt[n]{a} \cdot \sqrt[n]{b} = \sqrt[n]{ab} \). This equation represents a property of radicals that states that the product of two nth roots is equal to the nth root of the product of the two numbers, valid for any nonzero real numbers \(a\) and \(b\).

Step 2: Confirm the Conditions

We need to confirm the condition that both \(a\) and \(b\) are nonzero real numbers. Since the problem statement already specifies that we assume radicals represent nonzero real numbers, this condition is satisfied.

Step 3: Apply the Property of Radicals

According to the property of radicals, \(\sqrt[n]{a} \cdot \sqrt[n]{b} = \sqrt[n]{ab}\), as long as \(a\) and \(b\) are non-negative real numbers or \(n\) is an odd integer. Given the assumption that our radicals are real and non-zero, the expression holds true.

Key Concepts

Nonzero Real Numbers

Mathematically, a nonzero real number is any number on the real number line that is not equal to zero. The real numbers include both positive and negative numbers as well as changes fractions or irrational numbers, which include numbers like \ \(\pi\ \) and \ \(\sqrt{2}\ \). Real numbers without zero are particularly important in radical expressions because division by zero or taking the even root of zero leads to undefined expressions or discontinuities.
When working with equations involving radicals, it is crucial to ensure that all numbers involved are nonzero real numbers to allow the property of radicals to be applied accurately. Zero can drastically affect the validity of many mathematical expressions and properties. By restricting radicals to nonzero real numbers, we avoid these pitfalls and guarantee smoother calculations, including division and radical manipulations.

Nth Root

The nth root of a number represents a value that, when raised to the power of \ \(n\ \), gives the original number back. For instance, \ \(\sqrt[n]{a}\ \) seeks a number which when raised to the \ \(n\ \)th power equals \ \(a\ \). Common roots include the square root (2nd root) and the cube root (3rd root), but nth roots apply to any integer \ \(n\ \).
Considerations when working with nth roots depend on whether \ \(n\ \) is even or odd:

• If \ \(n\ \) is even, the number under the radical must be non-negative for real results.
• If \ \(n\ \) is odd, the number under the radical can be positive, negative, or zero.

Applying these considerations ensures correctness and helps prevent imaginary or undefined results in calculations.

Product of Radicals

The rule for the product of radicals states \ \(\sqrt[n]{a} \cdot \sqrt[n]{b} = \sqrt[n]{ab}\ \). This property allows the multiplication of numbers inside a radical directly, simplifying complex expressions.
This property can only be used under certain conditions:

• The radicands \ \(a\ \) and \ \(b\ \) must be non-negative if \ \(n\ \) is even; otherwise, the radicals would not be real numbers.
• The value of \ \(n\ \) must be consistent across all radicals being multiplied.

This powerful property can significantly reduce computation by allowing one to combine and simplify expressions having the same root, provided the constraints involving nonzero real numbers and \ \(n\ \) as established are followed.