Problem 19 - Elementary Algebra

Question

Make use of the power rule for quotients, the power rule for products, the power rule for powers, or a combination of these rules to simplify each expression. $$ \left[\frac{5 x^{4}(y+1)}{5 x^{4}(y+1)}\right]^{6} $$

Step-by-Step Solution

Answer

Answer: The simplified form of the given expression is 1.

Step 1: Simplify the fraction inside the brackets

We need to simplify the fraction inside the brackets: $$\frac{5x^4(y+1)}{5x^4(y+1)}$$ Since the numerator and denominator are the same, the fraction simplifies to 1: $$1$$

Step 2: Apply the power rule for quotients

Now, we need to apply the power rule for quotients to the simplified expression inside the brackets: $$[1]^6$$ Raising 1 to any power will result in 1. Therefore, the simplified expression is: $$1$$

Key Concepts

Simplifying Algebraic Expressions

One of the most frequent tasks in algebra is simplifying algebraic expressions. It involves reducing an expression to its simplest form while maintaining its value. This process can make calculations easier and help you understand the structure of a problem better.

In our exercise, we began by looking at a fraction with the same expression in the numerator and the denominator. When you encounter such a scenario, remember that any expression divided by itself equals 1, as long as it's not zero. This is a crucial concept in algebra because it simplifies what might look complex at first glance into something very manageable. Factoring, canceling common factors, and recognizing identity elements like 1 for multiplication are vital skills here. While on the surface it might seem like a small detail, this understanding is a key step in mastering algebra.

Exponent Rules

Exponents signify repeated multiplication and can markedly shorten how we write down large numbers or repeated calculations. There are several rules we use to work with exponents effectively, and one of the most important is the power rule. The power rule for quotients comes into play when you deal with a quotient, or a fraction, raised to an exponent. The rule states that you can distribute the exponent to both the numerator and the denominator of the fraction.

In the original exercise, however, the fraction simplified to 1 before we applied any exponent rules. But knowing that \( (a^m) ^ n = a^{m \times n} \) can be a powerful tool in your algebra arsenal. It's important to be able to identify when these rules apply to a problem, as they can significantly simplify and expedite your calculations.

Algebraic Fractions

Algebraic fractions are just like traditional fractions, but they contain algebraic expressions in the numerator and/or denominator. Just like with numerical fractions, our first move in simplifying algebraic fractions is usually to reduce them to their simplest form. This might involve factors canceling each other out or applying concepts such as the greatest common factor.

In our given problem, the algebraic fraction was simplified because the same expression was present in both the numerator and the denominator—it was already in its simplest form. Understanding how to work with algebraic fractions is crucial, as they often appear in calculus, equation solving, and problem modeling. Always look for opportunities to simplify algebraic fractions, as this can reveal the structure of the problem you're working with and, as seen in our exercise, can sometimes drastically simplify the entire expression.