Problem 74 - Precalculus: Functions and Graphs

Question

Simplify the expression, and rationalize the denominator when appropriate. $$\sqrt[5]{\frac{3 x^{11} y^{3}}{9 x^{2}}}$$

Step-by-Step Solution

Answer

\(\frac{x^{\frac{9}{5}} y^{\frac{3}{5}} \cdot 3^{4/5}}{3}\)

Step 1: Simplify Inside the Radical

First, simplify the expression inside the fifth root. The original expression is \[ \sqrt[5]{\frac{3 x^{11} y^{3}}{9 x^{2}}} \].Dividing the constants, we have \( \frac{3}{9} = \frac{1}{3} \).For the variables, divide \(x^{11}\) by \(x^{2}\):\[x^{11} \div x^{2} = x^{11-2} = x^{9}.\]The expression now simplifies to:\[ \sqrt[5]{\frac{1 \cdot x^{9} y^{3}}{3}}. \]

Step 2: Simplify the Fraction

The expression can be written as:\[ \frac{\sqrt[5]{x^{9}y^{3}}}{\sqrt[5]{3}}. \]Since \frac{x^{9}y^{3}}{3}\ is already simplified, we examine what can be further simplified in the radicals if needed, but here these are simplest forms.

Step 3: Apply the Fifth Root

To simplify \( \sqrt[5]{x^{9}y^{3}} \):\[ \sqrt[5]{x^{9}} = x^{\frac{9}{5}},\]\[ \sqrt[5]{y^{3}} = y^{\frac{3}{5}}.\]Combine these results:\[ \sqrt[5]{x^{9}y^{3}} = x^{\frac{9}{5}} y^{\frac{3}{5}}.\]

Step 4: Rationalize the Denominator

We have:\[ \frac{x^{\frac{9}{5}} y^{\frac{3}{5}}}{\sqrt[5]{3}}.\]To rationalize, multiply both numerator and denominator by \sqrt[5]{3^4}\ :\[ \left(\frac{x^{\frac{9}{5}} y^{\frac{3}{5}}}{\sqrt[5]{3}}\right) \cdot \left(\frac{\sqrt[5]{3^4}}{\sqrt[5]{3^4}}\right) = \frac{x^{\frac{9}{5}} y^{\frac{3}{5}} \cdot \sqrt[5]{81}}{3}. \]This results in:\[ \frac{x^{\frac{9}{5}} y^{\frac{3}{5}} \cdot 3^{4/5}}{3}. \]Further simplification is unnecessary.

Key Concepts

Simplifying Radicals

Simplifying radicals involves finding a more manageable form of the expression inside the radical. This step makes it easier to solve or further manipulate an expression. In the case of the expression \(\sqrt[5]{\frac{3x^{11}y^{3}}{9x^{2}}}\), we begin by simplifying the terms inside the fifth root:

• Start with the constant: \(\frac{3}{9} = \frac{1}{3}\).
• For the variables: divide \(x^{11}\) by \(x^{2}\). This results in \(x^{9}\).

Putting it all together, the simplified expression inside the radical is \(\sqrt[5]{\frac{x^9 y^3}{3}}\). This simplification helps maintain clarity, allowing further operations, like applying the root, to be executed more smoothly. Remember to always look for common factors and use the properties of exponents to simplify the radicals effectively.

Rationalizing the Denominator

Rationalizing the denominator is a process used to eliminate radicals from the bottom of a fraction. This makes the expression easier to handle and often brings it into a standard form. To rationalize the denominator of \(\frac{x^{\frac{9}{5}} y^{\frac{3}{5}}}{\sqrt[5]{3}}\), multiply both the numerator and the denominator by \(\sqrt[5]{3^4}\). This operation leads to:

• The numerator becomes \(x^{\frac{9}{5}} y^{\frac{3}{5}} \cdot \sqrt[5]{81}\).
• The denominator becomes \(3\) because \(\sqrt[5]{3} \cdot \sqrt[5]{3^4} = 3\).

The expression now looks like \(\frac{x^{\frac{9}{5}} y^{\frac{3}{5}} \cdot 3^{\frac{4}{5}}}{3}\). By rationalizing the denominator, the operation creates a neater expression where the radical has been effectively managed.

Fifth Roots

Fifth roots are a specific type of radical expression where the root is indexed by five. This means you're looking for a value which, when raised to the power of five, gives the original number or expression inside the radical. In our example, we find the fifth root of \(x^9 y^3\) as follows:

• The fifth root of \(x^9\) is expressed as \(x^{\frac{9}{5}}\).
• The fifth root of \(y^3\) is expressed as \(y^{\frac{3}{5}}\).

Combining these, \(\sqrt[5]{x^9 y^3}\) becomes \(x^{\frac{9}{5}} y^{\frac{3}{5}}\). Understanding fifth roots involves recognizing that fractional exponents relate directly to the index of the root, allowing the transformation between these two forms.

Fractional Exponents

Fractional exponents are another way to write roots; they provide a powerful shortcut method to solve problems involving radicals. The general rule states \(x^{m/n}\) means the \(n\)-th root of \(x^m\).In our expression \(x^{\frac{9}{5}} y^{\frac{3}{5}}\):

• The term \(x^{\frac{9}{5}}\) implies taking the fifth root of \(x^9\).
• Similarly, \(y^{\frac{3}{5}}\) implies the fifth root of \(y^3\).

Using fractional exponents, we avoid cumbersome radical notation, making calculations tidier. Additionally, fractional exponents adhere to the same rules as traditional exponents—enabling operations like multiplication, division, and simplification—so they blend seamlessly into algebraic expressions.