Problem 38 - Algebra A Combined Function

Question

Use the power rule and the power of a product or quotient rule to simplify each expression. $$ (a b)^{6} $$

Step-by-Step Solution

Answer

The simplified expression is \(a^6 b^6\).

Step 1: Understand the Expression

The given expression is \((ab)^6\), which means we need to raise the product \((ab)\) to the sixth power. We will apply properties of exponents to simplify this.

Step 2: Apply the Power of a Product Rule

The power of a product rule states that \((xy)^n = x^n \, y^n\). Using this rule, we rewrite \((ab)^6\) as \(a^6 \, b^6\).

Step 3: Final Simplified Expression

No further simplification is needed since both \(a^6\) and \(b^6\) are in their simplest forms. Thus, the simplified expression is \(a^6 \, b^6\).

Key Concepts

Power Rule

The power rule is an essential concept in algebra that helps simplify expressions with exponents. It's all about performing arithmetic operations with powers in a systematic way. When you raise a term to another power, you multiply the exponents together. This means if you have an expression like \(x^m\) raised to the power \(n\), you can express it as \(x^{m \cdot n}\). For example, \( (x^2)^3 = x^{2 \cdot 3} = x^6 \).

• Remember: Exponents tell us how many times to multiply a number by itself.
• The power rule simplifies computations, allowing us to manage larger exponents easily.

Using the power rule isn't just a shortcut; it's a valuable tool for simplifying expressions with multiple levels of exponents.

Power of a Product Rule

The power of a product rule is a handy tool when dealing with expressions that involve multiple factors raised to an exponent. It states that \( (xy)^n = x^n \, y^n \). This rule tells us that to raise a product of numbers to a power, we can raise each factor to that power separately.

For example, consider \( (3b)^4 \). Using the power of a product rule, it turns into \( 3^4 \, b^4 \). This rule simplifies complex expressions and makes it easier to handle larger calculations.

• Apply it by distributing the exponent to each factor within the parentheses.
• This rule is particularly useful when working with algebraic expressions containing variables.

Knowing how to apply this rule efficiently can streamline your problem-solving process.

Exponents

Exponents are a fundamental part of algebra that indicate how many times a number, called the base, is multiplied by itself. In the expression \( x^n \), \( x \) is the base, and \( n \) is the exponent, also known as the power.

Key points about exponents include:

• An exponent of 1 means the number is itself: \( x^1 = x \).
• An exponent of 0 results in 1, for non-zero bases: \( x^0 = 1 \).
• Negative exponents indicate reciprocals: \( x^{-n} = \frac{1}{x^n} \).

Understanding these properties of exponents is essential when simplifying expressions. They dictate how numbers grow exponentially and allow us to manage calculations involving large numbers more efficiently.

Simplification

Simplification in mathematics involves reducing an expression to its most compact and manageable form. While simplifying, you follow a variety of rules, including those for exponents, to avoid unnecessary complexity.

Here are some important points about simplification:

• Combine like terms and use arithmetic operations to make the expression simpler.
• Apply all relevant rules, such as the power rule and power of a product rule, to break down complex expressions.
• Ensure your final expression has no like terms or extraneous factors lingering.

The goal is to make an expression as clear and concise as possible, so it's easier to understand and work with. Simplification is an integral part of solving algebraic problems, enabling you to focus on the core components of an expression.