Problem 4 - Elementary Algebra

Question

Multiply. (Assume all denominators are nonzero.) $$ 16 a 47 b 2 \cdot 49 b 32 a 3 $$

Step-by-Step Solution

Answer

The product is \( 4704a^2b^2 \).

Step 1: Rearrange the Terms

Reorder the terms in each product so that like terms are grouped together. This will simplify multiplication.Original term:\( 16a \cdot 47b \cdot 2 \) becomes \( 16 \cdot 2 \cdot a \cdot 47 \cdot b \).Original term:\( 49b \cdot 32a \cdot 3 \) becomes \( 49 \cdot 3 \cdot b \cdot 32 \cdot a \).

Step 2: Multiply the Coefficients

Identify the numeric coefficients in each reordered term and multiply them together.For \( 16a \cdot 47b \cdot 2 \): Multiply \( 16 \cdot 2 \), which equals \( 32 \).For \( 49b \cdot 32a \cdot 3 \):Multiply \( 49 \cdot 3 \), which equals \( 147 \).

Step 3: Combine Like Variables

Combine the variables in each product.For \( 32 \cdot a \cdot 47 \cdot b \), it remains as \( 32ab \).For \( 147 \cdot b \cdot 32 \cdot a \), it remains as \( 147ab \).

Step 4: Perform the Final Multiplication

Multiply the simplified numeric coefficients and variables from both products.Multiply the coefficients:\( 32 \cdot 147 = 4704 \).The variables multiply as follows:\( a \cdot b \cdot a \cdot b = a^2b^2 \).Thus, the result is \( 4704a^2b^2 \).

Key Concepts

Rearranging Terms

When dealing with algebraic expressions, rearranging terms can make complex multiplications seem simpler. The core idea is to group similar types of terms together. In our exercise, we have two expressions:

• \(16a \cdot 47b \cdot 2\)
• \(49b \cdot 32a \cdot 3\)

Rather than multiply them as they appear, rearranging allows us to combine like terms effortlessly. In the first expression, change it to \(16 \cdot 2 \cdot a \cdot 47 \cdot b\). For the second, reorder it to \(49 \cdot 3 \cdot b \cdot 32 \cdot a\). This step ensures easier multiplication in subsequent steps by putting numeric coefficients together and aligning variables.

Multiplication of Coefficients

Once you've rearranged the terms, the next step focuses on multiplying the coefficients. Coefficients are the numerical parts of the terms that directly multiply with the variables. In our example:

• From \(16 \cdot 2\), multiply to get \(32\).
• For \(49 \cdot 3\), the result is \(147\).

By isolating and multiplying the coefficients on their own, we simplify the product before introducing the variables. Always tackle coefficients first as this reduces the complexity when variables differ or have multiple exponents.

Combining Like Variables

In algebra, after dealing with coefficients, we must approach variables systematically by combining them when they are similar. Variables in our exercise are organized in the rearranged products, making them easier to multiply:

• For \(a \cdot 47 \cdot b\) in the first expression, it naturally forms \(32ab\).
• The second part, \(b \cdot 32 \cdot a\) simplifies to \(147ab\).

These steps keep variables in check by ensuring that identical variables were placed together in a commutative manner. This is crucial before multiplying the entire product to ensure clarity and accuracy.

Variables Multiplication

The final multiplication step involves both the coefficients and the variables. Here, we multiply everything together to obtain the final algebraic expression.

• Insert the multiplied coefficients: \(32 \cdot 147\) results in \(4704\).
• Multiply the variables: \(a \cdot b \cdot a \cdot b\) becomes \(a^2b^2\).

Thus, combining everything gives the result \(4704a^2b^2\), showing the power of breaking down the problem step-by-step. This approach ensures accuracy and helps visualize the interplay between numbers and variables in an organized manner.