Problem 45 - Elementary and Intermediate Algebra

Question

Perform the operations and simplify, if possible. See Example 5 $$\frac{10 a^{2}}{3 b^{4}} \cdot \frac{12 b^{3}}{5 a^{2}}$$

Step-by-Step Solution

Answer

The simplified expression is \( \frac{8}{b} \).

Step 1: Multiply Numerators

To multiply fractions, first multiply the numerators. We have \(10a^2\) and \(12b^3\). Multiply these together: \[ 10a^2 imes 12b^3 = 120a^2b^3 \]

Step 2: Multiply Denominators

Next, multiply the denominators \(3b^4\) and \(5a^2\). Do this by: \[ 3b^4 imes 5a^2 = 15a^2b^4 \]

Step 3: Formulate the New Fraction

Combine the results from steps 1 and 2 to form the new fraction:\[ \frac{120a^2b^3}{15a^2b^4} \]

Step 4: Simplify the Fraction

To simplify, first note that \(a^2\) is in both the numerator and the denominator, and can cancel out. Also, \(b^3\) cancels with part of \(b^4\), leaving \(b\) in the denominator. What remains is:\[ \frac{120}{15b} \]

Step 5: Simplify the Numerical Coefficient

Divide the numerical coefficient \(120\) by \(15\) to simplify:\[ \frac{120}{15} = 8 \]Thus, the complete simplified expression is:\[ \frac{8}{b} \]

Key Concepts

Multiplying Fractions

Multiplying fractions might feel daunting, but it's all about keeping things orderly. When multiplying two fractions, you multiply the numerators together and the denominators together to form a new fraction.

In our example,

• The numerators were \( 10a^2 \) and \( 12b^3 \).
• Multiplying them gives \( 120a^2b^3 \).
• On the denominator side, we had \( 3b^4 \) and \( 5a^2 \).
• Multiplying these equals \( 15a^2b^4 \).

Remember, focus on one step at a time. Keep multiplying straight across and avoid handling the fractions any further until you get a single, combined result. Once you've done the top and bottom separately, you have a new fraction formed by these results.

Canceling Common Factors

After multiplying, you'll find that your new fraction might look complex. The next step is to simplify by canceling common factors.

Common factors appear in both the numerator and the denominator. In fraction notation, these can be canceled out, as they evenly divide into both parts. In the example:

• Notice \( a^2 \) appears in both the numerator and denominator, so it cancels out.
• The term \( b^3 \) in the numerator can be canceled with \( b^3 \) of the \( b^4 \) in the denominator, leaving \( b \) in the denominator.

Focus on identifying terms or numbers that repeat in both parts of the fraction. Canceling doesn't change the value of the expression but makes it easier to manage.

Simplifying Expressions

Once you've canceled out all the common factors, what's left is a simpler fraction. You may still need to simplify any remaining numbers.

In our scenario, we ended up with the expression \( \frac{120}{15b} \). This step involves simplifying the numerical coefficient, or the large numbers, that you might still have. Here's how:

• Divide 120 by 15, resulting in 8.

Your finalized fraction will now be \( \frac{8}{b} \). This simplified expression is much more manageable and easier to understand.

Simplifying fractions is all about finding the simplest form to convey the same value, which allows for easier interpretation and further mathematical operations.