Problem 72 - Precalculus: Functions and Graphs

Question

Simplify the expression, and rationalize the denominator when appropriate. $$\sqrt[4]{\frac{x^{7} y^{12}}{125 x}}$$

Step-by-Step Solution

Answer

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

Step 1: Simplify Inside the Radical

Start by simplifying the expression inside the fourth root. The expression inside is \( \frac{x^7 y^{12}}{125 x} \). Divide \( x^7 \) by \( x \) to get \( x^6\). The expression becomes: \( \frac{x^6 y^{12}}{125} \).

Step 2: Apply Fourth Root

Now apply the fourth root to each term separately. For \( x^6 \), you have \( \sqrt[4]{x^6} = x^{6/4} = x^{3/2} \). For \( y^{12} \), \( \sqrt[4]{y^{12}} = y^{12/4} = y^3 \). And for \( 125 \), \( \sqrt[4]{125} = 5^{3/4} \).

Step 3: Rationalize the Denominator

Since \( 5^{3/4} \) is not rational, we need to rationalize. Multiply numerator and denominator by \( 5^{1/4} \) to convert the denominator to 5. The expression becomes: \( \frac{x^{3/2} y^3 \cdot 5^{1/4}}{5} \).

Step 4: Multiply and Simplify

Finally, multiply the expressions in the numerator. You multiply \( x^{3/2} y^3 \) by \( 5^{1/4} \), resulting in \( x^{3/2} y^3 5^{1/4} \). The simplified expression is: \( \frac{x^{3/2} y^3 5^{1/4}}{5} \).

Key Concepts

Simplifying Radical Expressions

When working with radical expressions, it is important to start by simplifying the terms within the radical. This means looking for ways to reduce the expression inside the radical sign. Simplifying radical expressions often involves:

• Cancelling common factors in the numerator and the denominator.
• Reducing powers of like bases.

In this case, the initial expression is \( \sqrt[4]{\frac{x^{7} y^{12}}{125 x}} \). To simplify inside the radical, divide \( x^7 \) by \( x \), leaving \( x^6 \). This results in the simplified expression inside the fourth root: \( \frac{x^6 y^{12}}{125} \).
Whenever you see terms that can be reduced, always handle them first because it makes further calculations easier and less prone to error.

Fourth Root

The fourth root of a number or an expression is essentially the number that, when multiplied by itself four times, yields the original number. Let's look at how the fourth root is applied to each component.
The fourth root \( \sqrt[4]{x^6} \) turns into \( x^{6/4} \), which simplifies to \( x^{3/2} \) using exponent rules.

• For \( y^{12} \), the fourth root is \( y^{12/4} = y^3 \).
• For 125, express it as \( 5^3 \), so \( \sqrt[4]{125} = 5^{3/4} \).

Finding the fourth root often turns complex expressions into simpler terms, making them easier to work with in further calculations. Always handle each term separately and use simple fractions to express the exponents.

Exponent Rules

Exponent rules are key to solving many mathematical expressions, especially when dealing with radical expressions. These rules include:

• Power Rule: \( (a^m)^n = a^{m\cdot n} \)
• Product Rule: \( a^m \cdot a^n = a^{m+n} \)
• Quotient Rule: \( \frac{a^m}{a^n} = a^{m-n} \) where \( n < m \)

In this problem, after applying the power rule to turn \( x^6 \) into \( x^{6/4} \), you then simplify the fraction to \( x^{3/2} \). Similarly, \( y^{12/4} \) simplifies to \( y^3 \).
Finally, rationalizing the denominator makes use of these rules as you manipulate the terms to eliminate radicals in the denominators. Multiplying the numerator and the denominator by \( 5^{1/4} \) helps convert \( 5^{3/4} \) into a whole number to rationalize the denominator effectively. Understanding and correctly applying these rules can simplify the process and lead to the correct solution.