Problem 85 - Intermediate Algebra

Question

Use rational expressions to write as a single radical expression. $$ \sqrt[3]{y} \cdot \sqrt[5]{y^{2}} $$

Step-by-Step Solution

Answer

\( \sqrt[3]{y} \cdot \sqrt[5]{y^2} = \sqrt[15]{y^{11}} \)

Step 1: Express each radical using rational exponents

The expression given is \( \sqrt[3]{y} \cdot \sqrt[5]{y^{2}} \). We first convert each radical into an expression with a rational exponent. For \( \sqrt[3]{y} \), this becomes \( y^{1/3} \), and for \( \sqrt[5]{y^{2}} \), this becomes \( y^{2/5} \). Thus, the expression is now \( y^{1/3} \cdot y^{2/5} \).

Step 2: Use the property of exponents to combine terms

Recall the property \( a^m \cdot a^n = a^{m+n} \). Applying this property, we combine the terms: \( y^{1/3} \cdot y^{2/5} = y^{1/3 + 2/5} \).

Step 3: Simplify the exponent expression

To add the exponents \( \frac{1}{3} \) and \( \frac{2}{5} \), we find a common denominator. The least common denominator of 3 and 5 is 15. Convert \( \frac{1}{3} = \frac{5}{15} \) and \( \frac{2}{5} = \frac{6}{15} \). Add these to get \( \frac{5}{15} + \frac{6}{15} = \frac{11}{15} \). Thus, the expression is \( y^{11/15} \).

Step 4: Convert back to radical notation

Finally, convert the expression back from a rational exponent to radical notation. \( y^{11/15} \) can be written as \( \sqrt[15]{y^{11}} \). Therefore, the expression \( \sqrt[3]{y} \cdot \sqrt[5]{y^{2}} \) simplifies to \( \sqrt[15]{y^{11}} \).

Key Concepts

Rational Exponents

Rational exponents are a way to express roots and powers in the same notation. They provide a convenient method to switch between radical forms and exponential forms. For example, instead of writing cube root \( \sqrt[3]{x} \), we can express it as \( x^{1/3} \). A rational exponent is simply a fraction.

• The numerator indicates the power. In \( x^{m/n} \), the variable is raised to the power \( m \).
• The denominator indicates the root. \( n \) instructs us to take the \( n \)-th root of the base.

This conversion is particularly useful when dealing with multiple roots during calculations. Once expressions are converted to rational exponents, they can be manipulated much like other power and exponent expressions.

Properties of Exponents

Properties of exponents allow us to manipulate expressions involving powers efficiently. One important property is the product of powers property, \( a^m \cdot a^n = a^{m+n} \), which helps combine like bases. This is crucial when solving expressions with multiple radicals or exponents. Another useful property is the power of a power property, which states \((a^m)^n = a^{m\cdot n}\). When dealing with fractional exponents,

• The base stays the same.
• Add the exponents together when multiplying like bases. This means if you have \( y^{1/3} \cdot y^{2/5} \), you add the exponents (\( 1/3 \) and \( 2/5 \)) together.

Understanding these properties simplifies the process of working with complex expressions. This makes calculations quicker and more intuitive.

Simplifying Radical Expressions

Simplifying radical expressions is about making them as straightforward as possible while retaining their mathematical value. The goal is to express a complex radical in its simplest form. This often involves converting radicals to rational exponents, using exponent properties to simplify, and then converting back to radical form if needed.

• Start with converting radicals to rational exponents for ease of calculation.
• Use properties of exponents to add, subtract, or otherwise manipulate the exponents.
• Finally, if the problem requires, convert back to radical form for the simplest representation.

For example, given an expression like \( \sqrt[3]{y} \cdot \sqrt[5]{y^2} \), turn it into \( y^{1/3} \cdot y^{2/5} \).Then use rational exponents to simplify it to \( y^{11/15} \).Then, it can be expressed as \( \sqrt[15]{y^{11}} \), which is a simplified radical expression. The process requires careful application of exponent rules and often uses basic arithmetic skills for finding common denominators.