Problem 87 - Algebra A Combined Function

Question

Simplify each expression. $$ (4 a b)^{3} $$

Step-by-Step Solution

Answer

\(64a^3b^3\)

Step 1: Identify the Expression

We have the expression \((4ab)^3\). This means we need to cube the entire expression inside the parentheses.

Step 2: Apply the Power to Each Term

When raising a product to an exponent, you apply the exponent to each factor inside the parentheses. So, \((4ab)^3 = 4^3 \cdot a^3 \cdot b^3\).

Step 3: Calculate Each Term

Compute each term separately: - \(4^3 = 4 \times 4 \times 4 = 64\) - \(a^3\) remains \(a^3\)- \(b^3\) remains \(b^3\)

Step 4: Combine Terms

Combine the terms to get the final expression. So, \((4ab)^3 = 64a^3b^3\).

Key Concepts

Simplifying Expressions

Simplifying expressions is a key skill in algebra. It involves reducing an expression into its simplest form. This makes it easier to interpret and solve. Here, our task was to simplify the expression \((4ab)^3\).

• First, we identify that the expression needs to be raised to a power, meaning every part of it must be repeated a certain number of times.
• Next, we apply the exponent internally to each part of the expression. In our case, each factor inside the parentheses received the power of 3.
• It's important to distribute the exponent correctly to avoid errors.

If you can be systematic and careful in approaching each factor, simplifying becomes straightforward and less prone to mistakes.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and arithmetic operations. In our exercise, the expression \((4ab)^3\) consists of a number 4, and variables \(a\) and \(b\).

• Each part of the expression, like numbers and variables, play crucial roles.
• Variables like \(a\) and \(b\) help represent unknown values.
• We use algebraic rules to manipulate these expressions for simplification or solving.

The important aspect of working with algebraic expressions is recognizing how each component—numbers and letters, work together. This helps in performing operations like factoring, expanding, or simplifying.

Power of a Product

The power of a product rule is a fundamental rule in exponentiation. This rule states that when you have a product raised to an exponent, you apply the exponent to every factor within the parentheses. For the expression \((4ab)^3\), this means applying the power of 3 to each of \(4\), \(a\), and \(b\).

• Breaking down the expression using this rule: \((4ab)^3 = 4^3 \cdot a^3 \cdot b^3\).
• Firstly, compute for the numerical factor (\(4^3\)), which provides a concrete number in the expression.
• Next, simply apply the power to the variables, which often retain their form as \(a^3\) and \(b^3\).

The goal is to simplify without altering the value of the expression. This rule ensures that every component inside the product gets treated equally, achieving a complete simplification. Understanding and applying this principle is vital for handling more complex algebraic problems.