Problem 65 - Elementary Algebra

Question

Use the power rules for exponents to simplify the following problems. Assume that all bases are nonzero and that all variable exponents are natural numbers. $$ \left[\frac{x^{2}(y-1)^{3}}{(x+6)}\right]^{4} $$

Step-by-Step Solution

Answer

Question: Simplify the following expression using the power rules for exponents: \(\left(\frac{x^2(y-1)^3}{x+6}\right)^4\) Answer: \(\frac{x^8(y-1)^{12}}{(x+6)^{4}}\)

Step 1: Apply the power of 4 to the expression components

Apply the exponent of 4 to each part of the expression: \(x^2\), \((y-1)^3\), and \((x+6)\). This will result in \((x^2)^4\), \([(y-1)^3]^4\), and \([(x+6)]^4\).

Step 2: Apply the power rule to simplify the exponents

Multiplying the exponents, we get \(x^{2\cdot4}\), \((y-1)^{3\cdot4}\), and \((x+6)^4\) which simplifies to $$ x^{8}\frac{(y-1)^{12}}{(x+6)^{4}} $$

Step 3: Combine the terms to obtain the final simplified expression

Combine the terms from the previous step to have the final simplified expression: $$ \frac{x^8(y-1)^{12}}{(x+6)^{4}} $$This is the final answer.

Key Concepts

Simplifying Exponential Expressions

Dealing with exponential expressions often intimidates students, but understanding power rules can make the process much easier. Simplifying expressions like \(\left[\frac{x^{2}(y-1)^{3}}{(x+6)}\right]^{4}\) involves applying exponent laws to make the expression more manageable. The first step is to distribute the exponent outside the bracket across all terms inside it.

It's essential to remember that when you raise a power to a power, you multiply the exponents. For instance, \( (x^2)^4 \) becomes \( x^{2\cdot4} \) or \( x^8 \). This rule applies to every term with an exponent in the expression. Additionally, anything raised to the first power is itself, and any base raised to the zeroth power is one, assuming the base is not zero. These rules allow us to transform complicated expressions into simpler forms that are easier to work with.

Exponent Multiplication

When multiplying exponents with the same base, you add the exponents. However, in the context of the given exercise \(\left[\frac{x^{2}(y-1)^{3}}{(x+6)}\right]^{4}\), we are not directly multiplying exponents, but rather applying a power to an already exponentiated term, which requires us to multiply the exponents.

Rules of Exponent Multiplication

• If you have \( (a^m)^n \), then the result is \( a^{m \cdot n} \).
• For a term like \( (ab)^n \), distribute the exponent to get \( a^n b^n \).
• Division within a power, such as \( (\frac{a}{b})^n \), becomes \( \frac{a^n}{b^n} \).

Applying these rules, we multiply the exponents in the exercise to simplify the expression. This expedites evaluating algebraic expressions or solving equations using these principles.

Algebraic Expressions

Algebraic expressions are the cornerstone of algebra and consist of variables and constants combined with arithmetic operations. In the given exercise, we're working with a complex algebraic expression that includes exponents.

After applying the power rules for exponents, we end up with single terms raised to powers, such as \( x^8 \) and \( (y-1)^{12} \) and a polynomial \( (x+6)^4 \). The final step is to write these together in simplified form, ensuring the terms are correctly organized according to the laws of exponents.

The simplification of algebraic expressions can be furthered by factoring, combining like terms, or expanding products of binomials, though such steps are beyond the scope of this exercise. Understanding how to simplify expressions is vital for solving equations, understanding functions, and analyzing graphs in more advanced areas of mathematics.