Problem 90 - Algebra and Trigonometry

Question

In Exercises 83–90, perform the indicated operation or operations. $$ \frac{(5 x-3)^{6}}{(5 x-3)^{4}} $$

Step-by-Step Solution

Answer

After simplifying, the result of the given exercise will be \( (5x-3)^2 \).

Step 1 Identify like bases

Recognize that the numerator and denominator of the given fraction contain like bases, which is \( (5x - 3) \). Both the numerator and the denominator have exponents.

Step 2 Apply the exponent rule

Apply the rule of exponents for division, which states that when we divide expressions with the same base, we subtract the exponent of the denominator from the exponent of the numerator: \( (a^m/a^n) = a^{(m-n)} \).

Step 3 Perform the operation

Subtract the exponent 4 of the denominator from the exponent 6 of the numerator to obtain : \( (5x-3)^{(6-4)} \).

Key Concepts

Like Bases in Algebra

Understanding 'like bases' is a fundamental concept in algebra that simplifies the process of manipulating expressions. Like bases occur when expressions have the same base but are raised to different powers, for example, \(5x - 3\) in the numerator and \(5x - 3\) in the denominator. When faced with like bases, we're able to apply specific exponent rules to simplify complex-looking expressions. Algebraic fractions with like bases can often be simplified by using exponent laws, reducing the complexity of the problem.

Essentially, recognizing like bases allows you to see past the intimidating notation and spot opportunities for simplification. It's similar to realizing that you have the same type of currency in several denominations, which can be more efficiently combined or exchanged when you identify them as the same basic unit.

Exponents Division

The rule for division of exponents with like bases is a powerful shortcut in algebra. When you have an expression where a common base is raised to a power and is being divided by the same base raised to another power, you don't need to expand the expression fully. Instead, you can subtract the exponents. For instance, when you see \(a^m / a^n\), you can simplify this to \(a^{m-n}\).

This rule is particularly useful because it can drastically reduce the complexity of the algebra involved. The step of subtracting exponents is akin to reducing a fraction to its simplest form—you're trimming down the information to the most compact, efficient representation. More than just a mechanical rule, it's an example of how algebra seeks to make relationships between numbers more understandable and manageable.

Simplifying Algebraic Fractions

Simplifying algebraic fractions is similar to simplifying numerical fractions; it involves reducing the expression to its least complex form. This usually entails canceling out common factors between the numerator and the denominator, which often include like bases with exponents. In our example \(\frac{(5x-3)^6}{(5x-3)^4}\), simplification involves applying the exponents division rule. Algebraic fractions can often appear daunting, but breaking them down using exponent rules and identifying like terms can make the process more approachable.

Simplification isn't just cosmetic; it can reveal insights into the nature of the algebraic relationship and is essential for further operations like solving equations or graphing functions. Moreover, a simplified algebraic fraction is easier to evaluate, differentiate, or integrate, which becomes helpful in more advanced stages of mathematics.