Problem 33 - Elementary and Intermediate Algebra

Question

Use the product rule for exponents to simplify each expression. Write the results using exponents. $$ \left(a^{2} b^{3}\right)\left(a^{3} b^{3}\right) $$

Step-by-Step Solution

Answer

The simplified expression is \( a^5 b^6 \).

Step 1: Identify Each Base

We have the expression \( \left(a^{2} b^{3}\right)\left(a^{3} b^{3}\right) \). Notice that this expression has two bases: \( a \) and \( b \). Each base is raised to a power in both parts of the expression.

Step 2: Apply the Product Rule for Exponents

The product rule for exponents states that \( x^{m} \cdot x^{n} = x^{m+n} \). Apply this rule separately for each base:\[ (a^2)(a^3) = a^{2+3} = a^5 \]\[ (b^3)(b^3) = b^{3+3} = b^6 \]

Step 3: Combine the Results

Now, bring together the simplified expressions for each base. The expression simplifies to:\( a^5 b^6 \).

Key Concepts

Simplifying Expressions

Simplifying expressions is like tidying up a messy room. You want everything to be neat and as simple as possible. When you simplify an expression, your goal is to make it shorter and easier to work with while keeping the same value.

Here, we simplify the expression \( \left(a^{2} b^{3}\right)\left(a^{3} b^{3}\right) \) by using the product rule for exponents. This rule helps us combine similar parts of the expression, reducing it to fewer terms. With simplifying, we aim to turn a complex problem into an easier one. It makes calculations quicker and helps us see patterns or solutions we might not notice otherwise.

So remember, simplifying isn't changing the expression's value, just making it clearer!

Base and Exponent Identification

Identifying the base and exponent in an expression is crucial. It’s like knowing the characters before you start a story. In the expression \( \left(a^{2} b^{3}\right)\left(a^{3} b^{3}\right) \), the bases are the "characters" we focus on: \( a \) and \( b \).
Each base is raised to a specific power, which is called an exponent. The exponent tells us how many times we multiply the base by itself.

For example:

• In \( a^2 \), \( a \) is the base and \( 2 \) is the exponent.
• In \( b^3 \), \( b \) is the base and \( 3 \) is the exponent.

When dealing with expressions that have exponents, always note the base and exponent first. This helps us apply rules like the product rule, to easily simplify expressions.

Combining Like Terms

Combining like terms is an effective method when you plot to simplify expressions. When we say "like terms," in the context of exponents, we're talking about terms that have the same base.
In the expression \( \left(a^{2} b^{3}\right)\left(a^{3} b^{3}\right) \), we combine terms with the same base using the exponents.

Here’s how it works:

• You see \( a^2 \) and \( a^3 \), which share the base \( a \).
• You also see \( b^3 \) and \( b^3 \), which share the base \( b \).

By applying the product rule for exponents, we add up the exponents for each base:

• \( a^{2} \cdot a^{3} = a^{2+3} = a^{5} \)
• \( b^{3} \cdot b^{3} = b^{3+3} = b^{6} \)

This process shows us how to neatly package these terms together, further simplifying the expression to \( a^{5} b^{6} \). Recognizing and combining like terms is a key skill in algebra!