Problem 11 - Algebra 2

Question

Simplify. Assume that no variable equals 0. $$ \left(\frac{1}{3} a^{8} b^{2}\right)\left(2 a^{2} b^{2}\right) $$

Step-by-Step Solution

Answer

\( \frac{2}{3} a^{10} b^4 \)

Step 1: Apply the Distributive Property

Multiply the coefficients of the expressions: \( \frac{1}{3} \times 2 = \frac{2}{3} \).

Step 2: Multiply the Like Bases

For the variable \(a\), use the property of exponents \(a^m \times a^n = a^{m+n}\): \(a^8 \times a^2 = a^{10}\).

Step 3: Multiply the Like Bases

For the variable \(b\), apply the same exponent rule: \(b^2 \times b^2 = b^{4}\).

Step 4: Combine the Results

Combine the results from each variable and the coefficients to get the simplified expression: \( \frac{2}{3} a^{10} b^4 \).

Key Concepts

Multiplication of Coefficients

When simplifying algebraic expressions, one of the first steps is to multiply coefficients. Coefficients are the numerical parts of terms in an expression. In our example, the coefficients are \( \frac{1}{3} \) and \( 2 \). To multiply them, simply perform the arithmetic operation: \( \frac{1}{3} \times 2 = \frac{2}{3} \).
This operation combines the numerical parts of the expressions, which is essential for simplifying the entire expression. Always ensure to multiply coefficients separately from variables.

Distributive Property

The distributive property is a crucial concept in algebra that states \( a(b + c) = ab + ac \). Although in this problem we don't see an obvious addition, we use this property when multiplying terms across parentheses.
It helps us handle multiplication involving multiple terms.

• Multiply each term within the parentheses by the outer term.
• This can apply to numbers, variables, or even more complex expressions.

In our exercise, after handling the coefficients with \( \frac{2}{3} \), we distribute this result to all similar bases, which sets the stage for simplifying the expression.

Properties of Exponents

Exponents represent repeated multiplication. Understanding properties of exponents is key to simplifying expressions with variables. For example, using the rule \( a^m \times a^n = a^{m+n} \), we combine powers with the same base.
In the given expression:

• For \( a \): \( a^8 \times a^2 = a^{8+2} \) resulting in \( a^{10} \).
• For \( b \): \( b^2 \times b^2 = b^{2+2} \) leading to \( b^4 \).

These properties allow us to consolidate expressions with similar bases efficiently.

Multiplying Like Bases

When multiplying like bases, focus on combining their exponents. "Like bases" means the variables are the same, like \( a \) with \( a \) or \( b \) with \( b \). In multiplication, add the exponents as per the exponent law mentioned above.
This method reduces multiple terms into a single term:

• If terms are like \( a^m \) and \( a^n \), the product is \( a^{m+n} \).
• For terms like \( b^p \) and \( b^q \), the product is \( b^{p+q} \).

This simplification makes the final expression more compact and manageable. For our example, this process transforms the terms to get to \( a^{10} \) and \( b^4 \), incorporating the coefficients to form \( \frac{2}{3} a^{10} b^4 \).