Problem 34 - A Graphical Approach to College Algebra

Question

If possible, simplify each radical expression. Assume that all variables represent positive real numbers. $$\sqrt[4]{\frac{3}{2}}$$

Step-by-Step Solution

Answer

\(\sqrt[4]{\frac{3}{2}}\) is already simplified.

Step 1: Understanding the Radical

We start with the expression \( \sqrt[4]{\frac{3}{2}} \). This is a fourth root of a fraction. Our goal is to understand the structure so we can proceed with simplification if possible.

Step 2: Using Rational Exponents

We can rewrite the fourth root in terms of rational exponents: \( \sqrt[4]{\frac{3}{2}} = \left(\frac{3}{2}\right)^{1/4} \). This representation might make it easier to work with the expression.

Step 3: Checking for Simplification

We check to see if \( \frac{3}{2} \) can be further simplified within the radical. Since both 3 and 2 are prime numbers, the expression \( \left(\frac{3}{2}\right)^{1/4} \) cannot be simplified further using factors.

Key Concepts

Simplifying expressions

Simplifying expressions can seem daunting at first, especially when they involve radicals or roots. The goal is to express the expression in its most compact and understandable form. Let's break this process down.

If you're given an expression with a radical, such as the fourth root, you're often able to use exponent rules to simplify it. The first step is to understand what is being asked. In our example, we're dealing with the fourth root of a fraction, \( \sqrt[4]{\frac{3}{2}} \).

A useful strategy when simplifying is to look for common factors or terms that can be simplified. However, it's not always possible. In cases where the numbers inside the radical are prime, like \( \frac{3}{2} \), the expression can't be simplified further when both 3 and 2 remain prime and don’t have common factors with the root index. The expression stays as it is, and we explore different forms like rational exponents to work with it.

Rational exponents

Working with rational exponents can often make solving or simplifying radical expressions easier. Rational exponents essentially transform a radical into an expression with an exponent.

For example, the radical \( \sqrt[4]{\frac{3}{2}} \) can be rewritten using rational exponents as \( \left(\frac{3}{2}\right)^{1/4} \). Here, the index of the root becomes the denominator of the exponent. So, a fourth root becomes an exponent of \( \frac{1}{4} \).

This representation can be easier to manipulate, especially when dealing with equations or attempting further simplification by aligning it with multiplication, division, or expression forms that include similar bases. Although in some cases, as our example shows, further simplification isn't possible, using rational exponents is a valuable tool in algebraic manipulation.

Fourth roots

Fourth roots are a type of radical expression where we seek a number that, when multiplied by itself four times, equals the given number. It's written as \( \sqrt[4]{a} \). Think of it as the opposite of raising to the power of four.

Understanding fourth roots becomes easier once you know that it's equivalent to raising the number to the power of \( \frac{1}{4} \), as shown in rational exponents: \( \left(\frac{3}{2}\right)^{1/4} \).

In practical applications, finding exact fourth roots can be challenging without a calculator unless dealing with perfect fourth powers (like 16, where \( \sqrt[4]{16} = 2 \)). In algebra, fourth roots often require similar strategies as simplifying square roots, looking for factors that might simplify the expression, although not always possible, just like in our given expression containing prime numbers.