Problem 39 - Elementary Algebra

Question

For the following problems, expand the quantities so that no exponents appear. $$ \left(9 a^{3} b^{2}\right)^{3} $$

Step-by-Step Solution

Answer

Answer: The expanded form of the expression is \(729a^9b^6\).

Step 1: Apply the power of a product rule

Applying the power of a product rule to the expression: \((9a^3b^2)^3 = 9^3(a^3)^3(b^2)^3\)

Step 2: Simplify the expression

Simplifying the exponentiations: \(9^3 = 729\) \(a^3^3 = a^{(3*3)} = a^9\) \(b^2^3 = b^{(2*3)} = b^6\)

Step 3: Combine the results

Now, combining all the simplified results to get the final expanded expression: \(729a^9b^6\)

Key Concepts

Power of a Product Rule

Understanding the power of a product rule is essential when working with expressions that contain both coefficients and variables raised to powers. This rule allows you to distribute the power to each factor inside a term individually when the whole term is raised to an exponent.

Let's take the expression \((9a^3b^2)^3\) as an example. Here, the entire term inside the parentheses is being raised to the third power. By applying the power of a product rule, we distribute the exponent of 3 to each individual part within the parentheses:

• The coefficient 9 is raised to the third power: \(9^3\).
• The variable \(a^3\) is raised to the third power: \((a^3)^3\).
• The variable \(b^2\) is raised to the third power: \((b^2)^3\).

This rule helps simplify the expression by breaking it down into manageable parts, making the underlying multiplication clearer. By understanding and applying this rule, you ensure a procedural approach to solving complex problems.

Simplifying Expressions

Simplifying expressions involves performing the calculations indicated by the exponentiations and organizing the expression into its simplest form. After applying the power of a product rule, the next logical step is to calculate the powers indicated.

Let's break it down further:

• Calculate \(9^3\). This means multiplying 9 by itself three times: \(9 \times 9 \times 9 = 729\).
• For \((a^3)^3\), you multiply the exponents: \(3 \times 3 = 9\), resulting in \(a^9\).
• Similarly, for \((b^2)^3\), multiply the exponents: \(2 \times 3 = 6\), which simplifies to \(b^6\).

After completing these calculations, you combine them to form the simplified expression, which in this case is \(729a^9b^6\). This step confirms that each component is correctly computed, eliminating any unnecessary complexity.

Algebraic Expansion

Algebraic expansion means expressing a term that includes an exponent into its fully expanded form, devoid of any further exponentiation. This is particularly useful when dealing with polynomial expressions.

In our given exercise, the expression \((9a^3b^2)^3\) is expanded by first using the power of a product rule and then simplifying step by step:

• Each element within the expression is accounted for by raising it to the 3rd power.
• After simplification, each element contributes to the final expanded form: 729 for the coefficient, \(a^9\) from the variable \(a^3\), and \(b^6\) from \(b^2\).

The end result, \(729a^9b^6\), is a full expansion of the original, showcasing the complete breakdown of each component of the term. Algebraic expansion, thus, enhances the understanding of how polynomial terms function together by reconfiguring terms into simpler, examinable forms.