Problem 76 - Algebra A Combined Function

Question

Use rational expressions to write as a single radical expression. $$ \frac{\sqrt[4]{a}}{\sqrt[5]{a}} $$

Step-by-Step Solution

Answer

The expression \( \frac{\sqrt[4]{a}}{\sqrt[5]{a}} \) simplifies to \( \sqrt[20]{a} \).

Step 1: Understanding the Problem

We are given two radicals, \( \sqrt[4]{a} \) and \( \sqrt[5]{a} \), and need to express them as a single radical expression. These radicals are part of a fraction.

Step 2: Convert Radicals to Exponents

The fourth root of \( a \) can be expressed as an exponent: \( a^{1/4} \). Similarly, the fifth root of \( a \) is \( a^{1/5} \). Therefore, we can rewrite the given expression \( \frac{\sqrt[4]{a}}{\sqrt[5]{a}} \) as \( \frac{a^{1/4}}{a^{1/5}} \).

Step 3: Apply the Quotient Rule for Exponents

Using the quotient rule for exponents \( \frac{x^m}{x^n} = x^{m-n} \), we can rewrite \( \frac{a^{1/4}}{a^{1/5}} \) as \( a^{1/4 - 1/5} \).

Step 4: Find a Common Denominator

To subtract the exponents \( \frac{1}{4} \) and \( \frac{1}{5} \), we need a common denominator, which is 20 in this case. Therefore, \( \frac{1}{4} = \frac{5}{20} \) and \( \frac{1}{5} = \frac{4}{20} \).

Step 5: Subtract the Exponents

Subtract the exponents: \( \frac{5}{20} - \frac{4}{20} = \frac{1}{20} \). So, \( \frac{a^{1/4}}{a^{1/5}} = a^{1/20} \).

Step 6: Convert Back to Radical Notation

Express \( a^{1/20} \) as a radical. This is the 20th root of \( a \), which is \( \sqrt[20]{a} \).

Key Concepts

Radical Expressions

Radical expressions involve roots of numbers or variables. For example, square roots and cube roots are common types of radical expressions. In mathematics, radicals are represented using symbols like \( \sqrt{} \), present in expressions like \( \sqrt{a} \) and higher roots like \( \sqrt[3]{b} \) for cube roots. When working with radicals, the number inside the root symbol is called the radicand, and the small number outside, the index, indicates the degree of the root.

Understanding how to manipulate radical expressions, like simplifying or combining them, is crucial for solving complex problems. In our exercise, we deal with the fourth and fifth roots of \( a \), represented as \( \sqrt[4]{a} \) and \( \sqrt[5]{a} \), respectively. Here, we learn to combine these radicals into a single expression.

Exponent Rules

Exponent rules are fundamental tools for simplifying expressions involving powers. They dictate how to handle expressions where numbers are repeatedly multiplied. Here are some key rules:

• The power of a product: \((xy)^n = x^n \cdot y^n\)
• The power of a power: \((x^m)^n = x^{mn}\)
• The power of a quotient: \((\frac{x}{y})^n = \frac{x^n}{y^n}\)

These rules help simplify expressions and solve equations efficiently.

In this exercise, exponent rules are used to convert radical expressions into a form with exponents. For instance, converting \( \sqrt[4]{a} \) into \( a^{1/4} \) by applying the conceptual relationship between radicals and exponents makes the problem easier to solve with further algebraic manipulation.

Quotient Rule for Exponents

The quotient rule for exponents is another powerful tool when simplifying expressions that involve fractions with the same base. This rule states: \( \frac{x^m}{x^n} = x^{m-n} \).

This means if you have a number raised to an exponent divided by the same number raised to another exponent, you subtract the exponents. It's beneficial when you're required to reduce expressions like \( \frac{a^{1/4}}{a^{1/5}} \). By applying the quotient rule, we simplify it to \( a^{1/4 - 1/5} \).

Once the exponents are subtracted by first finding a common denominator, the expression is reduced significantly, highlighting the utility of the quotient rule.

Converting Radicals to Exponents

Converting radicals to exponents transforms complex radical expressions into a more manageable form using fractional exponents. The general principle is that the \( n \)-th root of a number, \( \sqrt[n]{x} \), can be expressed as \( x^{1/n} \). This conversion facilitates easier calculation and manipulation using exponent rules.

• The square root becomes a \( \frac{1}{2} \) exponent: \( \sqrt{x} = x^{1/2} \)
• The cube root converts to a \( \frac{1}{3} \) exponent: \( \sqrt[3]{x} = x^{1/3} \)

In our exercise, \( \sqrt[4]{a} \) becomes \( a^{1/4} \) and \( \sqrt[5]{a} \) is \( a^{1/5} \). This conversion enables further manipulation, like combining or comparing these expressions using other exponent rules, outlined in previous sections.