Problem 81 - Algebra 2 and Trigonometry

Question

In \(74-82,\) write each expression as a power with positive exponents in simplest form. $$ \frac{\sqrt[5]{48 x y^{2}}}{\sqrt[3]{6 x^{2} y^{4}}} $$

Step-by-Step Solution

Answer

\( \frac{48^{\frac{1}{5}} \times 6^{-\frac{1}{3}}}{x^{\frac{7}{15}} \cdot y^{\frac{14}{15}}} \) where base terms follow exponent reduction rules.

Step 1: Express Roots as Exponents

Begin by rewriting the roots as fractional exponents. The fifth root of a number is the same as raising it to the power of \( \frac{1}{5} \), and the third root is the same as raising it to the power of \( \frac{1}{3} \). So, rewrite the expression as: \[\frac{(48xy^2)^{\frac{1}{5}}}{{(6x^2y^4)^{\frac{1}{3}}}}\]

Step 2: Distribute Exponents

Apply the properties of exponents by distributing the fractional exponents to each factor inside the parentheses: \[(48^{\frac{1}{5}} \cdot x^{\frac{1}{5}} \cdot y^{\frac{2}{5}}) \div (6^{\frac{1}{3}} \cdot x^{\frac{2}{3}} \cdot y^{\frac{4}{3}})\]

Step 3: Simplify Coefficients

Separate the coefficients and simplify them individually. Calculate \( 48^{\frac{1}{5}} \) and \( 6^{\frac{1}{3}} \): \[\frac{48^{\frac{1}{5}}}{6^{\frac{1}{3}}} = 48^{\frac{1}{5}} \cdot 6^{\frac{-1}{3}}\]Applying the property of exponents \(a^m \div b^n = \left( \frac{a}{b} \right)^m\), calculate the simplified expression.

Step 4: Use Properties of Exponents

Combine all terms using the properties of exponents, then rewrite the simplified expression with positive exponents: \[= (48^{\frac{1}{5}} \cdot 6^{\frac{-1}{3}}) \cdot (x^{\frac{1}{5} - \frac{2}{3}}) \cdot (y^{\frac{2}{5} - \frac{4}{3}})\]Calculate the expressions for \(x\) and \(y\): \[x^{\frac{1}{5} - \frac{2}{3}} = x^{\frac{3}{15} - \frac{10}{15}} = x^{-\frac{7}{15}}\]\[y^{\frac{2}{5} - \frac{4}{3}} = y^{\frac{6}{15} - \frac{20}{15}} = y^{-\frac{14}{15}}\]Therefore, \[= 48^{\frac{1}{5}} \cdot 6^{\frac{-1}{3}} \cdot x^{-\frac{7}{15}} \cdot y^{-\frac{14}{15}}\]

Step 5: Express Final Answer with Positive Exponents

To express the final answer with positive exponents, relocate any terms with negative exponents to the denominator: \[= \frac{48^{\frac{1}{5}} \cdot 6^{\frac{-1}{3}}}{x^{\frac{7}{15}} \cdot y^{\frac{14}{15}}}\]Convert the negatives into the denominator to get the simplest form as a fraction with positive exponents.

Key Concepts

Fractional Exponents

Fractional exponents might initially seem intimidating, but they offer a powerful way to deal with roots and powers simultaneously. When you see a root, like the fifth root or the third root, you can express these as powers with fractions:
\[ a^{\frac{1}{n}} \] where \( a \) is the base and \( n \) is the root. So, the fifth root, \( \sqrt[5]{a} \), is equivalent to \( a^{\frac{1}{5}} \), and the third root, \( \sqrt[3]{b} \), is \( b^{\frac{1}{3}} \). This transformation is incredibly useful as it allows us to use algebraic properties of exponents to manipulate these expressions more freely.

For example, in the exercise, the expression \( \frac{\sqrt[5]{48 x y^{2}}}{\sqrt[3]{6 x^{2} y^{4}}} \) was rewritten using fractional exponents as:
\[ \frac{(48xy^2)^{\frac{1}{5}}}{{(6x^2y^4)^{\frac{1}{3}}}} \] which provides a powerful framework for further simplification.

Exponent Properties

Exponent properties are like magic buttons for simplifying complex expressions. They allow us to handle even the trickiest of fractions and powers with ease. Here are some key properties to remember:

• The Power Rule: \( (a^m)^n = a^{m \times n} \) allows you to multiply the exponents together.
• The Quotient Rule: \( a^m \div a^n = a^{m-n} \) helps when dividing same bases with different exponents.
• The Product of Powers Rule: \( a^m \cdot a^n = a^{m+n} \), makes handling multiplication of the same base simpler.

In our expression, we used these properties to divide and distribute powers among the terms:
\[ (48^{\frac{1}{5}} \cdot x^{\frac{1}{5}} \cdot y^{\frac{2}{5}}) \div (6^{\frac{1}{3}} \cdot x^{\frac{2}{3}} \cdot y^{\frac{4}{3}}) \]
By applying these rules, we can manage and simplify otherwise complicated polynomial expressions efficiently.

Simplifying Expressions

Simplifying expressions is the art of making them easier to work with. Once you understand fractional exponents and exponent properties, simplifying any expression becomes a game of applying these rules and concepts thoughtfully.

Start by breaking down each part of the expression. Distribute exponents, as seen here, and simplify coefficients by calculating roots or fractional powers separately. For instance, simplifying \( 48^{\frac{1}{5}} \) and \( 6^{\frac{-1}{3}} \) gives us a more manageable expression. Next, you'll use subtraction rules on the exponents of like bases to simplify further:
\[ x^{\frac{1}{5} - \frac{2}{3}}, y^{\frac{2}{5} - \frac{4}{3}} \]
This results in negative exponents, which we convert to positive by moving them to the denominator:
\[ \frac{1}{x^{\frac{7}{15}} \, y^{\frac{14}{15}}} \]

Finally, simplify everything to its cleanest form using these steps. Practice regularly to get quicker and more comfortable with these transitions. Simplifying can be fun, you just need a little practice!