Problem 28 - Algebra A Combined Function

Question

Use the product rule to simplify each expression. $$ \left(a^{2} b\right)\left(a^{13} b^{17}\right) $$

Step-by-Step Solution

Answer

The expression simplifies to \( a^{15} b^{18} \).

Step 1: Identify the Rule

The exercise requires the use of the Product Rule of exponents. The Product Rule states that when multiplying two exponents with the same base, you add the exponents together. Mathematically, this can be written as \( a^m \cdot a^n = a^{m+n} \).

Step 2: Separate the Terms

Consider the expression \( \left(a^{2} b\right)\left(a^{13} b^{17}\right) \). Separate it into two parts: one for the base \( a \) terms and one for the base \( b \) terms.

Step 3: Apply the Product Rule to \( a \) Terms

For the base \( a \) terms, apply the Product Rule. Since both terms have base \( a \), use \( a^2 \cdot a^{13} = a^{2+13} = a^{15} \).

Step 4: Apply the Product Rule to \( b \) Terms

For the base \( b \) terms, apply the Product Rule. Since both terms have base \( b \), use \( b^1 \cdot b^{17} = b^{1+17} = b^{18} \).

Step 5: Combine Simplified Terms

Combine the results from Step 3 and Step 4. The expression simplifies to \( a^{15} b^{18} \).

Key Concepts

Product Rule

When you come across the product rule in exponents, it's like a magical tool for simplifying expressions. The rule is pretty straightforward: when you multiply two expressions with the same base, just add up the exponents. This is a lifesaver for simplification!

• Mathematically, it looks like this: \( a^m \cdot a^n = a^{m+n} \).
• This rule makes it super easy to deal with exponents in multiplication.

The key is making sure that the bases you are working with are the same. This makes it possible to just add up those exponents. It's a pretty universal tool and pops up a lot in algebra and beyond.

Simplifying Expressions

Once you understand the product rule, simplifying expressions becomes a breeze. Simplifying an expression means making it as neat as possible, often by combining like terms.
Let's look at our example from the exercise: \( (a^{2} b)(a^{13} b^{17}) \).

• First, identify similar bases and apply the product rule.
• Combine the terms by adding up the exponents of like bases.
• Remember, every time you simplify, you make the expression clearer and easier to understand.

The goal is to reach a simpler form, like turning "\( a^{2+13} \)" into "\( a^{15} \)." This is what makes working with algebraic expressions less intimidating and more fun!

Algebraic Expressions

An algebraic expression is like a sentence in mathematical language. It can include numbers, variables (like \( a \) and \( b \)), and operation signs. These expressions are everywhere in math!
Here's why they're important:

• They help us to generalize mathematical problems.
• They make it possible to solve problems without specific numbers, giving us solutions in terms of variables.
• When you learn to manipulate them, you unlock many secrets of algebra.

Every algebraic expression, like the one in our example \( (a^{2} b)(a^{13} b^{17}) \), represents a combination of variables and constants that can be simplified using rules like the product rule.

Multiplication of Exponents

Multiplication of exponents can seem tricky at first, but it's something you'll master with practice. This concept revolves around understanding how powers work when you're multiplying expressions.
For example, in \( a^2 \cdot a^{13} \), you're actually combining the exponents because they share the same base.

• Multiply bases by adding their exponents: \( a^m \cdot a^n = a^{m+n} \).
• This keeps things neat and the expressions easy to work with.
• Don't forget each base stands alone—only combine exponents when their bases match.

When mastered, this technique is powerful for managing complex algebraic expressions and making calculations much quicker.