Problem 537 - Elementary Algebra

Question

In the following exercises, simplify. (a) \(625^{\frac{1}{4}}\) (b) \(243^{\frac{1}{5}}\) (c) \(32^{\frac{1}{5}}\)

Step-by-Step Solution

Answer

(a) 5 (b) 3 (c) 2

Step 1: Simplify (a) \(625^{\frac{1}{4}}\)

First, factorize the base: \(625 = 5^4\). Then, apply the power rule, which states that \( (a^m)^n = a^{mn} \). Therefore, \( 625^{\frac{1}{4}} = (5^4)^{\frac{1}{4}} = 5^{4 \cdot \frac{1}{4}} = 5 \).

Step 2: Simplify (b) \(243^{\frac{1}{5}}\)

First, factorize the base: \(243 = 3^5\). Then, apply the power rule: \( (a^m)^n = a^{mn} \). Therefore, \( 243^{\frac{1}{5}} = (3^5)^{\frac{1}{5}} = 3^{5 \cdot \frac{1}{5}} = 3 \).

Step 3: Simplify (c) \(32^{\frac{1}{5}}\)

First, factorize the base: \(32 = 2^5\). Then, apply the power rule: \( (a^m)^n = a^{mn} \). Therefore, \( 32^{\frac{1}{5}} = (2^5)^{\frac{1}{5}} = 2^{5 \cdot \frac{1}{5}} = 2 \).

Key Concepts

Factorization

Factorization is the process of breaking down a number into its prime factors. This is a key step when working with exponents, especially when simplifying expressions. Prime factors are numbers that are only divisible by 1 and themselves.
For example, in the exercise, we need to simplify expressions with exponents. Let's consider the number 625 from part (a). We can rewrite it as a product of its prime factors:

• 625 = 5 x 5 x 5 x 5
• This is equivalent to 625 = 5^4

Once we have expressed the base as a power of its prime factor, simplifying the expression becomes much easier. Factorization is the foundation of many operations involving exponents. It allows us to rewrite complex expressions in a simpler form.

Power Rule

The power rule of exponents is a vital concept to master. It states that \[ (a^m)^n = a^{mn} \]. This rule helps simplify expressions where an exponent is raised to another exponent. Let's apply this to the examples from the exercise:

• Part (a): We found that 625 = 5^4. Now, using the power rule, we can simplify further: 625^{\frac{1}{4}} = (5^4)^{\frac{1}{4}} = 5^{4 \cdot \frac{1}{4}} = 5
• Part (b): For 243 = 3^5, applying the power rule: 243^{\frac{1}{5}} = (3^5)^{\frac{1}{5}} = 3^{5 \cdot \frac{1}{5}} = 3
• Part (c): With 32 = 2^5, using the power rule gives us: 32^{\frac{1}{5}} = (2^5)^{\frac{1}{5}} = 2^{5 \cdot \frac{1}{5}} = 2

Understanding and applying the power rule allows for efficient and accurate simplification of expressions involving exponents.

Exponents

Exponents represent repeated multiplication of a base number. They are written in the form \(a^b\), where \(a\) is the base and \(b\) is the exponent. In our exercise, understanding how to work with exponents is crucial.
When dealing with exponents, remember these key points:

• Multiplication: When multiplying similar bases, add the exponents: \(a^m \cdot a^n = a^{m+n}\).
• Division: When dividing similar bases, subtract the exponents: \(\frac{a^m}{a^n} = a^{m-n}\).
• Zero Exponent: Any non-zero base raised to the power of zero is 1: \(a^0 = 1\).
• Negative Exponent: A negative exponent means taking the reciprocal of the base and making the exponent positive: \(a^{-n} = \frac{1}{a^n}\).

Mastering the basic rules of exponents simplifies many mathematical operations and is foundational for more advanced math topics. In our exercise, recognizing and applying these rules helped us to smoothly simplify the given expressions.