Problem 16 - 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. $$ \left[\frac{2 a^{4}(b-1)}{3 b^{3}(c+6)}\right]^{4} $$

Step-by-Step Solution

Answer

Question: Simplify the expression $\left[\frac{2 a^{4}(b-1)}{3 b^{3}(c+6)}\right]^{4}$. Answer: $\frac{16 a^{16} (b-1)^4}{81 b^{12} (c+6)^4}$.

Step 1: Apply the power rule for quotients

The power rule for quotients states that \((\frac{a}{b})^n = \frac{a^n}{b^n}\). Applying this rule to the given expression, we have: $$ \left[\frac{2 a^{4}(b-1)}{3 b^{3}(c+6)}\right]^{4} = \frac{(2 a^{4}(b-1))^4}{(3 b^{3}(c+6))^4} $$

Step 2: Apply the power rule for products

The power rule for products states that \((ab)^n = a^n b^n\). Applying this rule to the numerator and denominator of the expression, we have: $$ \frac{(2 a^{4}(b-1))^4}{(3 b^{3}(c+6))^4} = \frac{2^4 a^{(4*4)} (b-1)^4}{3^4 b^{(3*4)} (c+6)^4} $$

Step 3: Simplify the expression

Now simplify the expression by calculating the powers: $$ \frac{2^4 a^{(4*4)} (b-1)^4}{3^4 b^{(3*4)} (c+6)^4} = \frac{16 a^{16} (b-1)^4}{81 b^{12} (c+6)^4} $$ So, the simplified expression is: $$ \frac{16 a^{16} (b-1)^4}{81 b^{12} (c+6)^4} $$

Key Concepts

Understanding the Power Rule for Quotients

The power rule for quotients is a useful tool when simplifying expressions involving division. It helps to raise both the numerator and the denominator to a given power. This can be understood with the formula: \(\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}\). This rule is applied when you have a fraction inside an exponent.

For instance, if you have an expression like \(\left(\frac{2a^4(b-1)}{3b^3(c+6)}\right)^4\), applying this rule allows you to separate the exponent across the fraction. Thus, you transform the expression into \(\frac{(2 a^{4}(b-1))^4}{(3 b^{3}(c+6))^4}\).

Without this rule, handling complex fractions would be cumbersome. By distributing the exponent across both the numerator and the denominator, we simplify calculations and clarity is improved.

Exploring the Power Rule for Products

The power rule for products is handy for simplifying expressions where several variables or terms are multiplied together, and we need to raise them to a power. The rule is expressed as \((ab)^n = a^n b^n\), meaning each factor in the product must be raised to the power individually.

Let's apply it to the expression from our example: \(\frac{(2 a^{4}(b-1))^4}{(3 b^{3}(c+6))^4}\). By applying the power rule for products, you split the exponents across each term within both the numerator and the denominator:

• Numerator: \((2 a^{4}(b-1))^4\) becomes \(2^4 a^{16} (b-1)^4\)
• Denominator: \((3 b^{3}(c+6))^4\) becomes \(3^4 b^{12} (c+6)^4\)

This process makes it significantly simpler to deal with the powers of each variable or number, allowing us to handle large and complex expressions with ease.

Mastering Simplifying Expressions

Simplifying expressions is an essential skill in algebra that involves reducing expressions to their simplest form. This often makes equations easier to read and solve. By using the power rules effectively, you can transform seemingly complicated expressions into much simpler ones.

In the final step of our example: \(\frac{2^4 a^{16} (b-1)^4}{3^4 b^{12} (c+6)^4}\), simplifying means carrying out the actual arithmetic for the numbers and reorganizing the expression clearly:

• \(2^4 = 16\)
• \(3^4 = 81\)

Thus, the expression simplifies to \(\frac{16 a^{16} (b-1)^4}{81 b^{12} (c+6)^4}\).

Simplifying brings the benefits of clarity and manageability to algebraic equations, making them less intimidating and more accessible for solving real-world problems.