Problem 13 - Elementary Algebra

Question

For the following problems, perform the multiplications and divisions. $$ \frac{12 a^{3}}{7} \cdot \frac{28}{15 a} $$

Step-by-Step Solution

Answer

Answer: The result of the multiplication is $\frac{21a^2}{15}$.

Step 1: Multiply the numerators

To multiply the numerators, we simply multiply the numbers and then multiply the variables. In this case, we have:$$ (12a^3) \cdot (28) = 12 \cdot 28 \cdot a^3 $$

Step 2: Multiply the denominators

To multiply the denominators, we simply multiply the numbers and then multiply the variables. In this case, we have:$$ (7) \cdot (15a) = 7 \cdot 15 \cdot a $$

Step 3: Simplify the fraction

Now we have:$$ \frac{12 \cdot 28 \cdot a^3}{7 \cdot 15 \cdot a} $$To simplify this fraction, we will first factor out any common factors between the numerators and the denominators. We can see that 12 and 28 have a common factor of 4, while 7 and 15 have no common factors; however, we do have a single factor of 'a' in both the numerator and the denominator. Thus, we simplify the fraction by dividing both the numerator and the denominator by their common factors:$$ \frac{12 \cdot 28 \cdot a^3}{7 \cdot 15 \cdot a} = \frac{3 \cdot 7 \cdot a^2}{1 \cdot 15} $$Finally, we multiply the remaining factors to get the simplified expression:$$ \frac{3 \cdot 7 \cdot a^2}{1 \cdot 15} = \frac{21a^2}{15} $$So, the result of the given multiplication is:$$ \frac{12a^3}{7} \cdot \frac{28}{15a} = \frac{21a^2}{15}

Key Concepts

Multiplication of Fractions

Multiplying fractions is more straightforward than you might think. Here’s how it works: You multiply the numerators (the top numbers) of the fractions together to get a new numerator. Similarly, multiply the denominators (the bottom numbers) of the fractions together to get a new denominator.

• Take the fractions \(\frac{12a^3}{7}\) and \(\frac{28}{15a}\).
• Multiply the numerators: \(12a^3 \times 28\).
• Multiply the denominators: \(7 \times 15a\).

What this does is condense the multiplication of two fractions into a single, larger fraction. Your resulting fraction before simplification looks like this: \(\frac{12 \cdot 28 \cdot a^3}{7 \cdot 15 \cdot a}\).
The multiplication of fractions is powerful because it allows you to combine multiple values into a single, simplified expression, which we’ll explore more in the next sections.

Division of Fractions

Dividing fractions might sound tricky, but it's quite easy once you remember a simple rule: to divide by a fraction, multiply by its reciprocal. The reciprocal of a fraction is just that fraction's numerator and denominator flipped.

• Given a division problem \(\frac{12a^3}{7} \div \frac{28}{15a}\), you invert the second fraction and multiply.
• This turns into \(\frac{12a^3}{7} \times \frac{15a}{28}\).

After changing a division into multiplication, follow the steps for multiplying fractions:

• Multiply the numerators: \(12a^3 \times 15a\).
• Multiply the denominators: \(7 \times 28\).

By following these rules, division of fractions can be managed easily as multiplication, providing similar results with fractions that can then be simplified.

Simplifying Fractions

Simplifying fractions comes after you multiply or divide them and is an essential step. It involves making the fraction as simple as possible by removing common factors from the numerator and the denominator. Here’s how:

• Look at your combined fraction \(\frac{12 \cdot 28 \cdot a^3}{7 \cdot 15 \cdot a}\).
• Identify the common factors in the numerator and denominator.
• Both 12 and 28 share a factor of 4; a common factor \(a\) is also found because of \(a^3\) and \(a\).

Dividing these common factors gives you a simplified form: \(\frac{3 \cdot 7 \cdot a^2}{1 \cdot 15}\).
From here, multiply the remaining numbers: \(\frac{21a^2}{15}\).
Simplification helps in reducing the fraction so it’s easier to understand or further manipulate in algebraic problems. Remember, a simpler form of a fraction is always more convenient and elegant.