Problem 41 - Algebra and Trigonometry

Question

\(29-46\) Simplify each expression. $$ \left(a^{2} a^{4}\right)^{3} $$

Step-by-Step Solution

Answer

\(a^{18}\)

Step 1: Understanding the Expression

The expression given is \((a^2 a^4)^3\). This involves simplifying exponents inside the parentheses and then applying an exponent rule to the whole expression.

Step 2: Simplify Inside the Parentheses

Combine the exponents inside the parentheses: \(a^2 \times a^4 = a^{2+4} = a^6\) using the rule of exponents \(a^m \times a^n = a^{m+n}\).

Step 3: Apply the Power of a Power Rule

Now apply the power of a power rule: \((a^6)^3\). According to the rule \((a^m)^n = a^{m \cdot n}\), we have \((a^6)^3 = a^{6 \cdot 3} = a^{18}\).

Step 4: Final Simplified Expression

The entire expression \((a^2 a^4)^3\) simplifies to \(a^{18}\).

Key Concepts

Power Rule

The power rule for exponents is a crucial concept in algebra. It states that when you raise a power to another power, you can multiply the exponents. This is expressed in the formula: \[ (a^m)^n = a^{m \cdot n} \]In our example, this rule is applied to \((a^6)^3\). Here, the base \(a^6\) is raised to the power of 3. According to the power rule, we multiply the exponents: \(6 \cdot 3\). As a result, the expression simplifies to \(a^{18}\).

• This rule helps simplify complex expressions, making algebra problems easier to manage.
• It reduces repeated multiplication into a single concise expression.

Understanding this rule is vital for solving problems involving nested exponents and can simplify calculations significantly.

Simplifying Expressions

Simplifying expressions involves reducing them to their most basic form. In the original exercise, simplifying starts inside the parentheses. You combine like terms and apply exponent rules to make the expression easier to evaluate.
For \( (a^2 a^4)^3 \), the terms \(a^2\) and \(a^4\) inside the parentheses are combined using the rule of exponents: \[ a^m \times a^n = a^{m+n} \]This simplification step turns \(a^2 a^4\) into \(a^6\).

• By simplifying expressions, you break down complex algebraic problems into smaller, more manageable parts.
• It reveals the essential components of the expression, making it easier to apply further rules.

The goal of simplifying is to make the expression concise and ready for evaluation or additional operations.

Algebraic Expressions

Algebraic expressions are compositions of numbers, variables, and operations that represent a value or set of values. They can range from simple, such as \(x + 2\), to more complex as seen in our example \((a^2 a^4)^3\).

• Variables act as placeholders and can represent any number. They are often combined with constants and coefficients.
• Operations in algebraic expressions include addition, subtraction, multiplication, division, and exponentiation.

Understanding how to manipulate algebraic expressions is essential in algebra. It requires knowledge of rules and properties, like associating, distributing, and especially the rules of exponents.
Working with these expressions involves:

• Identifying like terms and combining them
• Recognizing patterns and applying rules such as the power rule
• Simplifying to make the expression as straightforward as possible

Mastering algebraic expressions can help solve equations, model real-world scenarios, and develop a deeper understanding of mathematical relationships.