Problem 88 - Intermediate Algebra

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: Express Roots as Exponents

Rewrite the roots as exponents. The expression \( \sqrt[4]{a} \) is \( a^{1/4} \) and \( \sqrt[5]{a} \) is \( a^{1/5} \). So the expression becomes \( \frac{a^{1/4}}{a^{1/5}} \).

Step 2: Apply the Quotient Rule for Exponents

The quotient rule for exponents states \( \frac{a^m}{a^n} = a^{m-n} \). Apply this to \( \frac{a^{1/4}}{a^{1/5}} \), giving \( a^{1/4 - 1/5} \).

Step 3: Simplify the Exponent

Calculate \( 1/4 - 1/5 \). Find a common denominator for the fractions, which is 20. Rewrite as \( \frac{5}{20} - \frac{4}{20} = \frac{1}{20} \).

Step 4: Convert Back to Radical Form

Since \( a^{1/20} \) as a radical is \( \sqrt[20]{a} \), the single radical expression is \( \sqrt[20]{a} \).

Key Concepts

Radical Expressions

A radical expression involves a root, such as a square root or cube root. In our original exercise, we used a 4th root \( \sqrt[4]{a} \) and a 5th root \( \sqrt[5]{a} \). A radical involves two parts: the radicand and the index. The radicand is the number or expression inside the radical symbol, in this case, \( a \). The index is the small number outside the radical sign, indicating the degree of the root.

• 4th root \( \sqrt[4]{a} \) means a number when multiplied by itself four times equals \( a \).
• 5th root \( \sqrt[5]{a} \) means a number when multiplied by itself five times equals \( a \).

Understanding radical expressions is essential since they allow us to express numbers in a different form, often making complex calculations more manageable.

Exponents

Exponents are a shorthand way to represent repeated multiplication of the same number. The base is the number being multiplied, and the exponent is how many times the base is used in the multiplication.
For example, \( a^3 \) means \( a \times a \times a \). In the context of the exercise, we expressed roots as exponents:

• \( \sqrt[4]{a} \) becomes \( a^{1/4} \)
• \( \sqrt[5]{a} \) becomes \( a^{1/5} \)

This expression change is crucial because it allows us to apply exponent rules, such as the quotient rule, to manipulate and simplify expressions efficiently.

Quotient Rule for Exponents

The quotient rule for exponents is a fundamental property that allows for simplification of expressions where the same base number is divided. The rule states that when dividing exponents with the same base, you subtract the exponents: \[ \frac{a^m}{a^n} = a^{m-n} \] Applying this rule to the expression \( \frac{a^{1/4}}{a^{1/5}} \), we subtract the exponents: \( 1/4 - 1/5 = 1/20 \).
This calculation leads us to \( a^{1/20} \), simplifying the original expression through the use of this powerful rule. Understanding and applying the quotient rule simplifies expressions which can otherwise appear daunting.

Roots of Numbers

Roots of numbers are about finding a number which, when used in repeated multiplication, gives the original number. The index of the root tells us how many times the base number is multiplied by itself.
For instance, the square root \( \sqrt{a} \) finds a number which multiplied by itself gives \( a \). Similarly, a 20th root \( \sqrt[20]{a} \) finds a number which multiplied 20 times equals \( a \).
This concept is integral to converting back from exponents. When we found \( a^{1/20} \), it translates into the radical expression \( \sqrt[20]{a} \). By converting between roots and exponents, we gain flexibility in solving mathematical problems and ensure clarity in expressing complicated relationships.