Problem 9 - Elementary Algebra

Question

For the following problems, perform the multiplications and divisions. $$ \frac{4 a^{3}}{5 b} \cdot \frac{3 b}{2 a} $$

Step-by-Step Solution

Answer

Question: Multiply the fractions and express the result in the simplest form: $$\frac{4a^3}{5b} \cdot \frac{3b}{2a}$$ Answer: $$\frac{6a^2}{5}$$

Step 1: Rewrite the multiplication

First, we rewrite the multiplication of fractions as a single fraction with both numerators multiplied together and both denominators multiplied together: $$ \frac{4a^3}{5b} \cdot \frac{3b}{2a} = \frac{(4a^3)(3b)}{(5b)(2a)} $$

Step 2: Simplify the numerator and denominator

Next, we simplify the numerator and denominator by multiplying the terms inside the parentheses: $$ \frac{(4a^3)(3b)}{(5b)(2a)} = \frac{12a^3b}{10ab} $$

Step 3: Cancel out common terms

Now, we cancel out common terms from the numerator and the denominator. In this case, we can cancel \(a\) and \(b\) as well as the constants 2: $$ \frac{12a^3b}{10ab} = \frac{6a^2}{5} $$

Step 4: Write the final simplified expression

The final simplified expression is: $$ \frac{4a^3}{5b} \cdot \frac{3b}{2a} = \frac{6a^2}{5} $$

Key Concepts

Understanding Fractions

Fractions are a way of representing parts of a whole. In algebra, they work the same way as in arithmetic, but with variables and constants. Each fraction has:

• A numerator: the top part of the fraction.
• A denominator: the bottom part of the fraction.

When dealing with fractions in algebra, treat variables like numbers.
To multiply fractions, multiply the numerators together and the denominators together. This keeps the relationship between the parts intact and lets you simplify efficiently later.

Canceling in Algebraic Fractions

Canceling is simplifying a fraction by removing common factors from the numerator and the denominator. It makes the expression simpler and easier to manage.
In algebraic expressions:

• Look for common variables or constants in the numerator and denominator.
• Cancel out any factors that are the same.

This is similar to reducing fractions in basic math.
For example, in \( \frac{12a^3b}{10ab} \), both \( a \) and \( b \) can be canceled, as well as the number 2 from 12 and 10.
This simplification helps in performing operations accurately and quickly.

Multiplication and Division of Algebraic Expressions

When multiplying or dividing algebraic expressions, it's important to remember these steps:

• Multiply numerators together to get a new numerator.
• Multiply denominators together to get a new denominator.
• Simplify by canceling common factors if possible.

Division can be handled by flipping the second fraction and multiplying.
This means you take the reciprocal of the fraction that's being divided and then follow the multiplication steps.
Using these methods, the expression \( \frac{4a^3}{5b} \cdot \frac{3b}{2a} \) simplifies to \( \frac{6a^2}{5} \). By practicing multiplication and division of algebraic expressions, you become adept at simplifying complex problems.