Problem 98 - Algebra A Combined Function

Question

Simplify each expression. $$ \left(\frac{2 a b}{6 y z}\right)^{4} $$

Step-by-Step Solution

Answer

The expression simplifies to \( \frac{a^4b^4}{81y^4z^4} \).

Step 1: Rewrite the Expression with Exponentiation

The given expression is \( \left(\frac{2ab}{6yz}\right)^4 \). This means applying the exponent of 4 to both the numerator and the denominator separately. Rewrite it as \( \frac{(2ab)^4}{(6yz)^4} \).

Step 2: Apply the Exponent to the Numerator

Calculate \((2ab)^4\). This means raising each part inside the parentheses to the power of 4: \(2^4 \times a^4 \times b^4\). Calculate each part: \(2^4 = 16\), leaving \(a^4\) and \(b^4\). Hence, \((2ab)^4 = 16a^4b^4\).

Step 3: Apply the Exponent to the Denominator

Now compute \((6yz)^4\), which means \(6^4 \times y^4 \times z^4\). Calculate \(6^4\): \(6^4 = 1296\), and leave the variables as \(y^4\) and \(z^4\). Thus, \((6yz)^4 = 1296y^4z^4\).

Step 4: Form the Fraction with Simplified Powers

Use the results from Step 2 and Step 3 to form the complete expression: \( \frac{16a^4b^4}{1296y^4z^4} \).

Step 5: Simplify the Fraction

To simplify \(\frac{16a^4b^4}{1296y^4z^4}\), find the greatest common divisor of the numbers 16 and 1296, which is 16. Divide both the numerator and the denominator by 16 to get \( \frac{a^4b^4}{81y^4z^4} \). There are no common factors left to further simplify.

Key Concepts

Exponentiation

Exponentiation refers to the mathematical process of raising a number, known as the base, to a certain power, which is called the exponent. This operation is represented as \( a^n \), where \( a \) is the base, and \( n \) is the exponent. In simple terms, exponentiation tells us how many times to multiply a number by itself.
For example, in our exercise, we applied an exponent of 4 to both the numerator and the denominator. This meant multiplying the entire expression within the parentheses by itself four times.
Breaking it down, when raising an expression that includes several terms, like \((2ab)^4\), each component within the parentheses needs to be raised to the power of 4. Therefore, we get \(2^4\), \(a^4\), and \(b^4\). This step is crucial as it sets the foundation for simplifying complex algebraic expressions.

Rational Expressions

A rational expression is simply a fraction that involves polynomials in the numerator, the denominator, or both. It's similar to an ordinary fraction but with one key difference: its components are polynomials rather than simple numbers.
In the context of our problem, the expression \(\left(\frac{2ab}{6yz}\right)^4\) is a rational expression. Here, both the numerator and the denominator consist of algebraic terms. Understanding rational expressions is essential in algebra as they frequently appear in equations and functions.
When working with rational expressions, it's important to consider the rules of fractions, such as simplifying by canceling common factors or finding a common denominator when adding or subtracting.

Simplification

Simplification in algebra involves rewriting expressions in the most concise and manageable form possible. This often means reducing fractions, combining like terms, or factoring polynomials.
For this exercise, simplification involved applying the exponent to both the numerator and the denominator separately and then reducing the resulting fraction. After calculating \((2ab)^4 = 16a^4b^4\) and \((6yz)^4 = 1296y^4z^4\), we formed the expression as \(\frac{16a^4b^4}{1296y^4z^4}\).
The next step was to simplify this fraction by identifying and dividing both numerator and denominator by their greatest common divisor, which simplified the fraction to \(\frac{a^4b^4}{81y^4z^4}\). Simplification helps in making complex algebraic expressions more approachable and easier to work with.

Greatest Common Divisor

The greatest common divisor (GCD) is the largest number that divides two or more numbers without leaving a remainder. It's a crucial concept in mathematics, especially in simplifying fractions.
In this exercise, after obtaining the expression \(\frac{16a^4b^4}{1296y^4z^4}\), we needed to simplify it. We found that the GCD of the numbers 16 and 1296 was 16.
Dividing the entire fraction by this GCD resulted in a more simplified form: \(\frac{a^4b^4}{81y^4z^4}\). By correctly applying the GCD, we could reduce the fraction to its simplest terms, eliminating any unnecessary complexity. Understanding how to find and use the GCD is essential for efficiently simplifying fractions in algebra.