Problem 83 - Algebra A Combined Function

Question

Use rational expressions to write as a single radical expression. $$ \sqrt[5]{7} \cdot \sqrt[3]{y} $$

Step-by-Step Solution

Answer

\( \sqrt[15]{7^3 \cdot y^5} \)

Step 1: Convert Radicals to Exponents

The first step involves rewriting each radical expression in terms of exponents. Recall that the nth root of a number can be expressed as that number raised to the power of \( \frac{1}{n} \). So, rewrite \( \sqrt[5]{7} \) as \( 7^{\frac{1}{5}} \) and \( \sqrt[3]{y} \) as \( y^{\frac{1}{3}} \).

Step 2: Multiply the Expressions

Since both expressions are now in exponential form, multiply them together: \( 7^{\frac{1}{5}} \times y^{\frac{1}{3}} \). In exponentiation, multiplication of bases with different exponents keeps the bases separate and maintains their respective exponents.

Step 3: Re-compose as a Single Radical Expression

To express the resulting product as a single radical, recognize that a common expression with a single radical has a unified base. Combine the product under a single radical using the least common multiple (LCM) of the denominators of the fractional exponents. The LCM of 5 and 3 is 15. Thus express the radicals as a single 15th root: \( \sqrt[15]{7^3 \cdot y^5} \).

Key Concepts

Radicals

Radicals represent the root of a number. They are written using the radical symbol (√). For example, the radical expression \( \sqrt[5]{7} \) is the fifth root of 7. Radicals are a central concept in algebra and are vital for simplifying complex mathematical expressions. Handling radicals involves understanding their properties:

• The nth root of a number means finding a number which, when raised to the nth power, gives you the original number.
• A radical can be converted to an exponent by expressing the nth root as a power of \( \frac{1}{n} \).

This conversion is crucial in higher-level mathematics, as it allows operations with exponents to simplify complex radical expressions. It helps us transition between radicals and rational expressions smoothly.

Exponents

Exponents are a way to represent repeated multiplication. An expression like \( x^n \) means "x multiplied by itself n times." Let's look at key points about exponents:

• Exponents can be fractions, which ties them to radicals. For instance, \( a^{\frac{1}{n}} \) represents the nth root of a.
• When multiplying expressions with the same base, you add the exponents: \( a^m \times a^n = a^{m+n} \).
• With different bases, each base keeps its own exponent when multiplied: \( a^m \times b^n = a^m b^n \).

In our problem, we converted radicals to exponents to simplify the expression for multiplication. For \( \sqrt[5]{7} \), we wrote it as \( 7^{\frac{1}{5}} \), turning a radical into a manageable exponent form.This transformation into exponents allows for the application of exponent rules, streamlining complex calculations.

Least Common Multiple

The Least Common Multiple (LCM) is a concept within number theory that involves finding the smallest number that two or more numbers can divide into without leaving a remainder. Finding the LCM is essential when working with fractional exponents.Here’s how to find the LCM:

• List the multiples of each number.
• Identify the smallest common multiple.

For example, when combining \( 7^{\frac{1}{5}} \) and \( y^{\frac{1}{3}} \), the LCM of 5 and 3 is 15. This allows us to create a unified radical expression and write a single expression under one radical sign: \( \sqrt[15]{7^3 \cdot y^5} \).Understanding LCM helps us combine expressions by smoothing out fractional parts, making it easier to operate within a unified mathematical structure.

Multiplying Expressions

Multiplying expressions involves more than just arithmetic; it's about combining terms and simplifying results. When multiplying expressions in exponential form, you might encounter several scenarios:

• Like bases: Add the exponents (e.g., \( x^a \times x^b = x^{a+b} \)).
• Unlike bases: Retain the respective bases and exponents (e.g., \( a^m \times b^n \)).
• Combine using common multiples for fractional exponents (as in creating a unified radical expression).

In our exercise, we multiplied \( 7^{\frac{1}{5}} \) by \( y^{\frac{1}{3}} \). Each base remains separate in the product, but we recognize their exponents' roles. Finding a shared radical using the LCM lets us elegantly combine them: from \( 7^{\frac{1}{5}} \times y^{\frac{1}{3}} \) to \( \sqrt[15]{7^3 \cdot y^5} \).Multiplying expressions with radicals involves skillfully using exponents' properties, forming a bridge towards simplifying disparate mathematical elements into cohesive, singular expressions.