Problem 91 - Intermediate Algebra

Question

Use rational expressions to write as a single radical expression. $$ \frac{\sqrt[3]{a^{2}}}{\sqrt[6]{a}} $$

Step-by-Step Solution

Answer

The expression simplifies to \(\sqrt{a}\).

Step 1: Convert Radicals to Exponents

We start by using the property that \(\sqrt[n]{x} = x^{1/n}\). Thus, we can rewrite the given radicals as exponents: \ \sqrt[3]{a^2}\ becomes \(a^{2/3}\) and \(\sqrt[6]{a}\) becomes \(a^{1/6}\). Our expression thus becomes: \ \frac{a^{2/3}}{a^{1/6}}\.

Step 2: Apply Division Rule for Exponents

The rule for dividing expressions with the same base is \(a^{m} / a^{n} = a^{m-n}\). Applying this: \ \frac{a^{2/3}}{a^{1/6}} = a^{2/3 - 1/6}\.

Step 3: Simplify the Exponent

To subtract the exponents, \(\frac{2}{3} - \frac{1}{6}\), we need a common denominator. The least common denominator for 3 and 6 is 6. Thus, convert \(\frac{2}{3}\) to \(\frac{4}{6}\) so that: \ \frac{4}{6} - \frac{1}{6} = \frac{3}{6} = \frac{1}{2}\.

Step 4: Convert Exponent Back to Radical Form

Now that we have simplified the exponent to \(a^{1/2}\), express it as a radical: \(a^{1/2} = \sqrt{a}\). Therefore, \ \frac{\sqrt[3]{a^2}}{\sqrt[6]{a}}\ can be expressed as the single radical \ \sqrt{a}\.

Key Concepts

Simplifying Expressions

Simplifying expressions is an important part of algebra, especially when dealing with more complex calculations. The concept of simplifying involves rewriting an expression in its most concise form while retaining the original value. This process often includes

• Combining like terms
• Rationalizing denominators
• Replacing complex radicals or exponents

Understanding and applying the properties of exponents and radicals is crucial. For instance, when you see a radical expression like \(\sqrt[n]{x}\), it can be rewritten in a simpler exponent form, \(x^{1/n}\).
In our specific problem, the given expression, \(\frac{\sqrt[3]{a^{2}}}{\sqrt[6]{a}}\), was simplified by converting the radicals into rational exponents. This makes it easier to handle as algebraic fractions. You're turning a more complex radical notation into something more standard and manageable.

Rational Exponents

Rational exponents allow you to express roots, such as square roots or cube roots, using powers of fractions. This is a powerful tool because it unifies the operations of rooting and raising to a power into one. For example, a square root, \(\sqrt{a}\), can be transformed into \(a^{1/2}\). Similarly, a cube root, \(\sqrt[3]{a}\), can be written as \(a^{1/3}\).
When working with rational exponents, you'll often use the property that

• \(\sqrt[n]{x} = x^{1/n}\)
• Combining them through multiplication, \((x^{a/b}) \cdot (x^{c/d}) = x^{(a/b) + (c/d)}\)
• Or division, \(\frac{x^{m}}{x^{n}} = x^{m-n}\)

This helps simplify expressions significantly. For example, in our exercise, converting the expressions \(\sqrt[3]{a^2}\) and \(\sqrt[6]{a}\) into rational exponents— \(a^{2/3}\) and \(a^{1/6}\) respectively—made it easier to simplify using the rules of division.

Algebraic Fractions

Algebraic fractions involve the division of algebraic expressions. Much like numerical fractions, these require a strong understanding of basic operations like addition, subtraction, multiplication, and division. When you are combining and simplifying algebraic fractions, it is similar to handling numbers, but you work with variables and their exponents instead.
A critical step in simplifying algebraic fractions is the use of exponent rules, especially when the expressions are in the form of powers. Here's what to remember:

• The division rule, \(\frac{a^m}{a^n} = a^{m-n}\), simplifies expressions with the same base.
• Finding a common denominator is essential when adding or subtracting fractions.
• Cancelling common factors in numerators and denominators helps simplify the expression.

In our scenario, dividing \(a^{2/3}\) by \(a^{1/6}\) was made straightforward by subtracting the exponents as per the division rule. Then, simplifying the resulting fraction of exponents led to the final expression, \(a^{1/2}\), which was easily converted back to a radical form, \(\sqrt{a}\). This demonstrates how understanding and applying algebraic fraction principles efficiently streamline simplifying complex expressions.