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Magazine Chapter 1: Problem 74
Simplify the expression, and rationalize the denominator when appropriate. $$\sqrt[5]{\frac{3 x^{11} y^{3}}{9 x^{2}}}$$
Short Answer
Expert verified
\(\frac{x^{\frac{9}{5}} y^{\frac{3}{5}} \cdot 3^{4/5}}{3}\)
Step by step solution
01
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:\[ \sqrt[5]{\frac{1 \cdot x^{9} y^{3}}{3}}. \]
02
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.
03
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:\[ \sqrt[5]{x^{9}y^{3}} = x^{\frac{9}{5}} y^{\frac{3}{5}}.\]
04
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.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
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 \(x^{9}\).
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 \(\sqrt[5]{3} \cdot \sqrt[5]{3^4} = 3\).
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 \(y^{\frac{3}{5}}\).
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 \(y^3\).
