Problem 51 - A Graphical Approach to College Algebra

Question

If possible, simplify each radical expression. Assume that all variables represent positive real numbers. $$\sqrt[4]{\frac{g^{3} h^{5}}{9 r^{6}}}$$

Step-by-Step Solution

Answer

\( \frac{g^{3/4} h^{5/4}}{\sqrt{3} r^{3/2}} \)

Step 1: Understand the Radical Expression

The expression given is \( \sqrt[4]{\frac{g^{3} h^{5}}{9 r^{6}}} \). This is a fourth root of a fraction involving variables within the radical. Our task is to simplify it.

Step 2: Simplify the Fraction Inside the Radical

Inside the fourth root, we have \( \frac{g^{3} h^{5}}{9 r^{6}} \). Note the coefficients and variables with exponents in both the numerator and the denominator.

Step 3: Apply Fourth Root to the Fraction

The expression can be rewritten as \( \left(\frac{g^{3} h^{5}}{9 r^{6}}\right)^{1/4} \). This representation allows us to apply the fourth root to both numerator and denominator separately.

Step 4: Break Down the Radicals in the Numerator

The fourth root of \( g^3 h^5 \) can be rewritten as \( (g^3)^{1/4} \times (h^5)^{1/4} = g^{3/4} \times h^{5/4} \).

Step 5: Simplify the Denominator

The fourth root of the denominator \( 9r^6 \) is broken down as \( (3^2)r^6 \). So it becomes \( (3^2 r^6)^{1/4} = 3^{2/4} \cdot r^{6/4} = 3^{1/2} \cdot r^{3/2} \).

Step 6: Simplify the Coefficient

Calculate \( 3^{1/2} \), which is the square root of 3, so it becomes \( \sqrt{3} \).

Step 7: Combine Results from Numerator and Denominator

Combine all parts: the numerator \( g^{3/4} \cdot h^{5/4} \) with the simplified denominator \( \sqrt{3} \cdot r^{3/2} \). The simplified expression is \( \frac{g^{3/4} h^{5/4}}{\sqrt{3} r^{3/2}} \).

Key Concepts

Fourth Root

Understanding the concept of a fourth root is crucial when simplifying radical expressions, especially those involving fractions with variables. A fourth root, denoted as \( \sqrt[4]{x} \), is essentially a number that, when multiplied by itself four times, equals the original value \( x \). This is an extension of the square root, which involves multiplying a number by itself twice.

When working with expressions like \( \sqrt[4]{\frac{g^{3} h^{5}}{9 r^{6}}} \), we look at the fourth root of both the numerator and the denominator. This allows us to break down complex terms and simplify step by step. In our case, once we represent the expression as \( \left(\frac{g^{3} h^{5}}{9 r^{6}}\right)^{1/4} \), it becomes easier to apply the fourth root to individual parts.

Breaking down each component individually unlocks the ability to handle complex exponents and resulting simplification more effectively, leveraging the rules of exponents for a straightforward simplification process.

Fraction Simplification

Fraction simplification is an important step in reducing the complexity of radical expressions, especially those involving multiple variables and exponents. When you see a fraction like \( \frac{g^{3} h^{5}}{9 r^{6}} \) inside a radical, our goal is to simplify both its numerator and denominator independently to make the overall expression more manageable.

Start with observing the layout:

• The numerator \( g^3 h^5 \) contains variables with exponents.
• The denominator 9 is a constant, equivalently \( 3^2 \), along with variable \( r^6 \).

Simplifying the fraction means reducing the coefficients and adjusting the terms using properties of exponents, making the next steps involving the fourth root straightforward.
Keep in mind that while the variables' powers remain untouched during the initial simplification, this step sets the foundation for breaking down terms using the fourth root and ultimately presenting a clean, simple expression.

Variables with Exponents

Working with variables that have exponents can often seem daunting, but by understanding the principles, it becomes a systematic process of simplification. In the expression \( \sqrt[4]{\frac{g^{3} h^{5}}{9 r^{6}}} \), variables with exponents play a dominant role.

To simplify, apply the rule of exponents: for any expression \( (x^a)^{1/n} = x^{a/n} \). This rule helps break down expressions like \( (g^3)^{1/4} \) and \( (h^5)^{1/4} \), resulting in \( g^{3/4} \) and \( h^{5/4} \).

In the denominator, the similar application transforms terms like \( (r^6)^{1/4} \) to \( r^{6/4} \) or further reduced to \( r^{3/2} \). This step-by-step breakdown via exponent rules ensures each variable's power is handled accurately, simplifying the expression to its most fundamental form. By consistently applying these rules, variables with exponents become less intimidating and more manageable in the context of radical expressions.