Problem 43 - College Algebra with Modeling and Visualization

Question

Exercises 35-44: Use the product rule to simplify. $$ \left(2 a^{3}\right)\left(b^{2}\right)\left(a^{-4}\right)\left(4 b^{-5}\right) $$

Step-by-Step Solution

Answer

The simplified expression is \( \frac{8}{a \cdot b^{3}} \).

Step 1: Rearrange the Expression

Start by organizing the expression in terms of like bases: \( \left(2a^{3}\right) \left(a^{-4}\right) \left(b^{2}\right) \left(4b^{-5}\right) \). This gives us the expression \( 2 \cdot 4 \cdot a^{3} \cdot a^{-4} \cdot b^{2} \cdot b^{-5} \).

Step 2: Simplify the Constants

Multiply the constant numbers together: \( 2 \cdot 4 = 8 \). So the expression becomes \( 8 \cdot a^{3} \cdot a^{-4} \cdot b^{2} \cdot b^{-5} \).

Step 3: Use the Product Rule for Exponents on 'a'

Apply the product rule for exponents for the variable \( a \): \( a^{3} \cdot a^{-4} = a^{3+(-4)} = a^{-1} \). Now the expression is \( 8 \cdot a^{-1} \cdot b^{2} \cdot b^{-5} \).

Step 4: Use the Product Rule for Exponents on 'b'

Apply the product rule for exponents for the variable \( b \): \( b^{2} \cdot b^{-5} = b^{2+(-5)} = b^{-3} \). Now the expression is \( 8 \cdot a^{-1} \cdot b^{-3} \).

Step 5: Write the Expression with Positive Exponents Only

To express with only positive exponents, rewrite \( a^{-1} \) and \( b^{-3} \) as \( \frac{1}{a} \) and \( \frac{1}{b^{3}} \). Thus, the expression simplifies to \( 8 \cdot \frac{1}{a} \cdot \frac{1}{b^{3}} = \frac{8}{a \cdot b^{3}} \).

Key Concepts

Simplifying Algebraic Expressions

Simplifying algebraic expressions is about making them easier to work with. This can involve combining like terms, reducing fractions, or simply rewriting them in a more intuitive form. The goal is always to express things as compactly and clearly as possible.
For example, in the expression given, \( \left(2 a^{3}\right)\left(b^{2}\right)\left(a^{-4}\right)\left(4 b^{-5}\right) \),we start by rearranging and grouping like terms, which helps organize our work.

• Identify and collect similar bases to make application of the product rule easier.
• Simplify any multiplication of constants first, as it involves simple arithmetic.

In this scenario, simplifying ensures that multiple entities like bases or constants aren't appearing more complicated than necessary. The final expression obtained, in simpler terms, is \( \frac{8}{a \cdot b^{3}} \), which doesn't contain unnecessary elements and is easy to understand.

Negative Exponents

Negative exponents can initially seem tricky, but they have a straightforward explanation. They indicate that a number should be taken as the reciprocal raised to the opposite positive power.
When simplifying expressions involving negative exponents:

• A base with a negative exponent, such as \( a^{-n} \), means \( \frac{1}{a^{n}} \).
• Similarly, \( b^{-m} \) becomes \( \frac{1}{b^{m}} \).

Using these rules helps convey the original value in a way that is often more useful and easier to work with in equations and further calculations.
In our example, transforming \( a^{-1} \) and \( b^{-3} \) into fractions yields a clearer, positive exponent-filled expression\( \frac{1}{a} \) and \( \frac{1}{b^{3}} \).This helps in creating a final compact and elegant simplification of the entire algebraic expression.

Multiplying Exponents

The product rule for exponents is a handy tool when multiplying powers with the same base. According to this rule, you simply add the exponents if the bases are identical.
Here's how it works:

• For any given same base \( a \), \( a^{m} \times a^{n} = a^{m+n} \)
• This rule simplifies the process of multiplication by replacing complex product operations with simpler addition of the exponents.

In our given exercise, this rule helped combine \( a^{3} \cdot a^{-4} \) into \( a^{-1} \), and \( b^{2} \cdot b^{-5} \) into \( b^{-3} \). The product rule aids in minimizing clutter and streamlining the expression, clearing the way for more straightforward application of arithmetic or algebraic operations. This is especially important when dealing with more complex expressions or when preparing expressions for calculus operations.