Problem 113 - Intermediate Algebra

Question

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

Step-by-Step Solution

Answer

True

Step 1: Understand the Given Expression

We are given \( \frac{\sqrt[n]{a}}{\sqrt[n]{b}} \) and need to determine if it is equal to \( \sqrt[n]{\frac{a}{b}} \). Both radicals are assumed to represent nonzero real numbers.

Step 2: Apply Radicals' Property

Using the property that \( \frac{\sqrt[n]{a}}{\sqrt[n]{b}} = \sqrt[n]{\frac{a}{b}} \) is valid under the assumption that \( n \) is a real number and both \( a \) and \( b \) are nonzero, we express this as a single radical: \( \sqrt[n]{a} = a^{1/n} \) and \( \sqrt[n]{b} = b^{1/n} \). Thus the left-hand side becomes \( \frac{a^{1/n}}{b^{1/n}} = (\frac{a}{b})^{1/n} = \sqrt[n]{\frac{a}{b}} \).

Step 3: Validate the Expression for General Cases

The expression holds because when each component of a radical is divided, the division can be combined under a single radical. For any nonzero real numbers \( a \), \( b \), and a real number \( n \), the equality \( \frac{\sqrt[n]{a}}{\sqrt[n]{b}} = \sqrt[n]{\frac{a}{b}} \) remains valid, confirming the property of radicals.

Key Concepts

Properties of Exponents

In mathematics, exponents allow us to express repeated multiplication compactly. Understanding the properties of exponents is essential when working with radical expressions. Let's dive deeper into these properties.

• Power of a Power: When you raise a power to another power, you multiply the exponents. For example, \( (a^m)^n = a^{m \times n} \).
• Product of Powers: This property states that when you multiply similar bases, you add the exponents: \( a^m \times a^n = a^{m+n} \).
• Quotient of Powers: When dividing like bases, subtract the exponents: \( \frac{a^m}{a^n} = a^{m-n}, aeq0 \).
• Negative Exponents: A negative exponent indicates reciprocal powers: \( a^{-m} = \frac{1}{a^m} \).

These properties are foundational for simplifying complex expressions, including those involving radicals. By expressing radicals in exponential form, such as \( \sqrt[n]{a} = a^{1/n} \), we can apply these rules efficiently to solve problems.

Division of Radicals

When dividing radical expressions, knowing how to manipulate and simplify them is crucial. The division of radicals can be simplified by combining them under a single radical sign, given certain conditions are met.For example, the expression \( \frac{\sqrt[n]{a}}{\sqrt[n]{b}} = \sqrt[n]{\frac{a}{b}} \) illustrates this principle perfectly. Here’s how it works:

• Both quantities under the radicals, \(a\) and \(b\), must be nonzero real numbers. This ensures the division is valid and the expression is defined.
• Both radicals should have the same index number \(n\). This allows us to switch between dividing the radicals separately or under a single radical without changing the expression’s value.

This property leverages the equivalence \( \sqrt[n]{a} = a^{1/n} \) and \( \sqrt[n]{b} = b^{1/n} \), enabling us to rewrite the division as \( (\frac{a}{b})^{1/n} \), which is indeed \( \sqrt[n]{\frac{a}{b}} \). Understanding this concept simplifies the approach to handling division within radical expressions.

Algebraic Properties of Radicals

Algebraic manipulation of radicals is a critical skill in solving complex mathematical problems. Radicals, like roots, often appear in algebraic expressions, requiring specific properties for simplification, addition, subtraction, multiplication, and division.Key algebraic properties of radicals include:

• Simplicity of Operations: You can only combine like radicals (radicals with the same index and radicand) using addition or subtraction. For instance, \( \sqrt{a} + \sqrt{a} = 2\sqrt{a} \), but \( \sqrt{a} + \sqrt{b} \) cannot be simplified further.
• Multiplicative Property: When multiplying radicals with the same index, their product can be combined under a single radical: \( \sqrt[n]{a} \times \sqrt[n]{b} = \sqrt[n]{a \times b} \).
• Rationalizing Denominators: Ensure expressions don’t leave radicals in the denominator. For example, to rationalize \( \frac{1}{\sqrt{a}} \), multiply by \( \frac{\sqrt{a}}{\sqrt{a}} \), resulting in \( \frac{\sqrt{a}}{a} \).
• Inverses and Exponents: Radicals can transform into exponents and vice versa, using the exponent rule \( \sqrt[n]{a} = a^{1/n} \), making the simplification process flexible and broadening solution strategies.

Applying these properties helps to readily simplify and accurately solve expressions involving radicals, ensuring that the intricacies of algebra become manageable and coherent.