Problem 71 - Precalculus: Functions and Graphs

Question

Simplify the expression, and rationalize the denominator when appropriate. $$\sqrt[4]{\frac{5 x^{x} y^{3}}{27 x^{2}}}$$

Step-by-Step Solution

Answer

The simplified expression is \(\frac{5^{1/4}x^{(x-2)/4}y^{3/4}3^{1/4}}{3}\).

Step 1: Simplify the Expression Inside the Radical

To simplify \( \sqrt[4]{\frac{5x^x y^3}{27x^2}} \), begin by dividing the powers of \( x \). The exponent of \( x \) inside the fraction is \( x - 2 \). This simplifies the expression inside the radical to \( \sqrt[4]{\frac{5x^{x-2}y^3}{27}} \).

Step 2: Apply the Fourth Root to the Fraction

Next, apply the fourth root separately to the numerator and the denominator of the fraction: \( \sqrt[4]{5x^{x-2}y^3} \) and \( \sqrt[4]{27} \).

Step 3: Simplify Numerator with Exponents

For \( \sqrt[4]{5x^{x-2}y^3} \), express each part under the fourth root as separate fourth roots: \( \sqrt[4]{5} \cdot \sqrt[4]{x^{x-2}} \cdot \sqrt[4]{y^3} \). This becomes \( 5^{1/4} \cdot x^{(x-2)/4} \cdot y^{3/4} \).

Step 4: Simplify Denominator with Exponents

For \( \sqrt[4]{27} \), write \( 27 = 3^3 \) so that \( \sqrt[4]{27} = (3^3)^{1/4} = 3^{3/4} \).

Step 5: Rationalize the Denominator if Needed

The simplified expression is \( \frac{5^{1/4}x^{(x-2)/4}y^{3/4}}{3^{3/4}} \). To rationalize the denominator, multiply numerator and denominator by \( 3^{1/4} \) to get \( \frac{5^{1/4}x^{(x-2)/4}y^{3/4} \cdot 3^{1/4}}{3} \). Simplifying gives \( \frac{5^{1/4}x^{(x-2)/4}y^{3/4}3^{1/4}}{3} \).

Key Concepts

Rationalizing the Denominator

Rationalizing the denominator is an essential step in simplifying radical expressions. It means transforming a fraction so the denominator no longer contains radicals. This process is particularly important because having a radical in the denominator is not considered proper mathematical form. Here’s how you can rationalize a denominator:

• Identify the radical term in the denominator.
• Multiply the numerator and the denominator by a value that will cancel the radical from the denominator.

In the given problem, after simplifying the values under the fourth root, we ended up with the expression \(\frac{5^{1/4}x^{(x-2)/4}y^{3/4}}{3^{3/4}}\). The denominator \(3^{3/4}\) is not rational. To rationalize, we multiply both the numerator and denominator by \(3^{1/4}\). This gives us a new denominator of \(3\), which is a rational number. Remember, multiplying by \(3^{1/4}\) is like multiplying by 1, so it doesn’t change the overall value of the expression. The resulting expression has a rationalized denominator and is simplified further.

Exponent Rules

Understanding exponent rules is key to simplifying radical expressions. Exponents manage repeated multiplication and are governed by several rules which can simplify complex expressions. Here are three important exponent rules used in this problem:

• Product of Powers: When multiplying like bases, add their exponents: \(a^m \cdot a^n = a^{m+n}\).
• Power of a Power: When raising a power to another power, multiply the exponents: \((a^m)^n = a^{m \cdot n}\).
• Quotient of Powers: When dividing like bases, subtract their exponents: \(\frac{a^m}{a^n} = a^{m-n}\).

These rules help simplify the expression inside the radical. For instance, dividing \(x^x\) by \(x^2\) gives \(x^{x-2}\), using the quotient of powers rule. Correct application of these rules is crucial for simplifying expressions accurately.

Fourth Roots

The concept of fourth roots involves finding a number that, when multiplied by itself three more times (i.e., a total of four times), results in the original number. Simplifying fourth roots can seem complex, but can be broken down into simpler components.

• Definition: The fourth root of a number \(a\) is written as \(\sqrt[4]{a}\) and is a number \(b\) such that \(b^4 = a\).
• Simplification: To simplify an expression involving fourth roots, separate the components under the radical and take each part’s fourth root separately. For example, \(\sqrt[4]{27}\) means finding a number which, when raised to the power of 4, gives \(27\).

In this exercise, we separate \(27\) into its factors, recognizing it as \(3^3\). This allows us to express \(\sqrt[4]{27}\) as \((3^3)^{1/4} = 3^{3/4}\). By breaking numbers and expressions into their factors or smaller parts, we can manage the calculations more effectively and see the solution path more clearly.