Problem 76 - Intermediate Algebra

Question

Use rational exponents to simplify each radical. Assume that all variables represent positive numbers. $$ \sqrt[4]{36} $$

Step-by-Step Solution

Answer

\( \sqrt{6} \)

Step 1: Express the Radical as a Power

To simplify the expression \( \sqrt[4]{36} \), first rewrite it using rational exponents. The expression can be written as \( 36^{1/4} \). This is because a radical \( \sqrt[n]{x} \) can be represented as \( x^{1/n} \).

Step 2: Prime Factorization of the Base

Find the prime factors of 36: \( 36 = 2^2 \times 3^2 \). This factorization will help us in simplifying the expression further.

Step 3: Apply the Exponent to Each Factor

Using the properties of exponents, apply the \(1/4\) exponent to each prime factor: \[ (2^2 \times 3^2)^{1/4} = 2^{2/4} \times 3^{2/4} \].

Step 4: Simplify the Exponents

Simplify the fractions in the exponents: \[ 2^{2/4} = 2^{1/2} \quad \text{and} \quad 3^{2/4} = 3^{1/2} \]. This results in the expression \[ 2^{1/2} \times 3^{1/2} \].

Step 5: Convert Back to Radical Form

Convert the simplified expression with rational exponents back to radicals: \[ 2^{1/2} \times 3^{1/2} = \sqrt{2} \times \sqrt{3} = \sqrt{6} \].

Key Concepts

Simplifying Radicals

Simplifying radicals is the process of making a radical expression more manageable by rationalizing it. A radical involves roots, like square roots or cube roots, often represented with the radical sign (\( \sqrt{} \)). In the example given, \( \sqrt[4]{36} \), the goal is to simplify it into a cleaner form without the radical, if possible.
Here's how it works:

• Start by expressing the radical as a rational exponent. For instance, \( \sqrt[4]{36} \) becomes \( 36^{1/4} \). This helps you in applying the rules of exponents more comfortably.
• With rational exponents in play, you can simplify the base, which is to say, reduce the larger radical expression into a simpler form using properties of exponents. In our example, simplifying takes this way: \( \sqrt{6} \), after thorough processing.

Understanding simplification is vital as it makes other math operations like addition and multiplication with radicals easier.

Prime Factorization

Prime factorization is the method of breaking down a number into its smallest multiplicative components—prime numbers.
When simplifying radicals, especially when converting them to rational exponents, prime factorization becomes tremendously useful. In our example:

• The number 36 was decomposed to reveal its prime factors: \( 36 = 2^2 \times 3^2 \).
• This breakdown is crucial as it allows us to tackle each prime factor individually when applying rational exponents.

By dismantling numbers into their primes, it simplifies the exponent calculation process considerably, enabling a straightforward path to simplifying expressions. By understanding which primes contribute to the original number, you can simplify radicals effectively.

Properties of Exponents

The properties of exponents are mathematical rules that govern how exponents function and interact with each other. They are essential when working with rational exponents and simplifying radical expressions.
A few key properties that help in this context are:

• The Power of a Power rule: which states \((a^m)^n = a^{m*n}\). This rule was applied when we took each prime factor (from the prime factorization step) and raised it to the power of \(1/4\).
• The Product of Powers rule: \(a^m \times a^n = a^{m+n}\). Although not used directly in the example, understanding this property can be useful when combining terms.
• Fractional exponents: make it feasible to translate between radical notation and exponent notation seamlessly, like when \(\sqrt[4]{36}\) became \(36^{1/4}\).

Understanding these properties is pivotal in maneuvering through problems involving exponents, especially when it's necessary to manipulate complex expressions involving radicals.