Problem 29 - Elementary Algebra

Question

For the following problems, perform the multiplications and divisions. $$ \frac{12 a^{2} b^{3}}{-5 x y^{4}} \div \frac{6 a^{2}}{15 x^{2}} $$

Step-by-Step Solution

Answer

Question: Simplify the given expression: \(\frac{12 a^{2} b^{3}}{-5 x y^{4}} \div \frac{6 a^{2}}{15 x^{2}}\) Answer: \(\frac{-30 b^3 x^2}{y^4}\)

Step 1: Determine the reciprocal of the second fraction

The reciprocal of a fraction is the result of swapping the numerator and the denominator. So, for our second fraction, which is \(\frac{6a^2}{15x^2}\), the reciprocal is \(\frac{15x^2}{6a^2}\).

Step 2: Multiply the first fraction with the reciprocal of the second fraction

Now we need to multiply the first fraction, \(\frac{12a^2b^3}{-5xy^4}\), by the reciprocal of the second fraction, \(\frac{15x^2}{6a^2}\). To do this, we multiply the numerators together and the denominators together: $$ \frac{12 a^{2} b^{3}}{-5 x y^{4}} \times \frac{15 x^{2}}{6 a^2} = \frac{12a^2b^3 \cdot 15x^2}{-5xy^4 \cdot 6a^2} $$

Step 3: Simplify the resulting expression

To simplify the resulting expression, we need to cancel out common factors in the numerator and denominator. Here are the common factors: - \(a^2\) in both the numerator and denominator - \(6\) (we can divide 12 and -5*6 by 6) Now let's cancel out these factors and simplify the expression: $$ \frac{12a^2b^3 \cdot 15x^2}{-5xy^4 \cdot 6a^2} = \frac{(2)(6)a^2b^3 \cdot (3)(5)x^2}{(5)(-1)x(6)y^4 \cdot a^2} = \frac{2 \cdot 3 \cdot 5 b^3 x^2}{(-1) y^4}= \frac{-30 b^3 x^2}{y^4} $$ So, the simplified expression is: $$ \frac{12 a^{2} b^{3}}{-5 x y^{4}} \div \frac{6 a^{2}}{15 x^{2}} = \frac{-30 b^3 x^2}{y^4} $$

Key Concepts

Reciprocal

Understanding the reciprocal of a number or a fraction is crucial in algebraic operations. The reciprocal of a fraction is obtained by swapping its numerator and denominator.
For example, given a fraction \ \( \frac{a}{b} \ \), its reciprocal will be \ \( \frac{b}{a} \ \). This concept is particularly important when dealing with division of fractions.

When you divide by a fraction, you essentially multiply by its reciprocal. This is why finding the reciprocal is the first step in division operations involving fractions. In our problem, the reciprocal of \( \frac{6a^2}{15x^2} \) is \( \frac{15x^2}{6a^2} \). By learning the reciprocal of fractions, you become capable of transitioning division problems into multiplication problems, simplifying the process greatly.

Multiplication of Fractions

Multiplying fractions involves a straightforward process that starts by multiplying the numerators together and the denominators together. Suppose you have two fractions \ \( \frac{a}{b} \) and \ \( \frac{c}{d} \).
Their product is calculated as follows: \ \( \frac{a}{b} \) x \ \( \frac{c}{d} = \frac{ac}{bd} \).

In our exercise, this concept comes into play after finding the reciprocal of the second fraction in the division problem. We use it to transform the division into a multiplication task: \( \frac{12 a^{2} b^{3}}{-5 x y^{4}} \times \frac{15 x^{2}}{6 a^2} \).

• Multiply the numerators: \(12a^2b^3 \cdot 15x^2 \)
• Multiply the denominators: \(-5xy^4 \cdot 6a^2 \)

This process is a standard approach and ensures that all terms are combined correctly within the resulting fraction.

Division of Fractions

To divide one fraction by another, we multiply the first fraction by the reciprocal of the second. This action is rooted in the principles of fractions, where division by a number is defined as multiplication by its reciprocal.
Let's consider \( \frac{12 a^{2} b^{3}}{-5 x y^{4}} \div \frac{6 a^{2}}{15 x^{2}} \).
We convert this to a multiplication problem:

• Find the reciprocal of \( \frac{6 a^{2}}{15 x^{2}} \), which is \( \frac{15 x^{2}}{6 a^{2}} \).
• Change the division sign to multiplication, resulting in: \( \frac{12 a^{2} b^{3}}{-5 x y^{4}} \times \frac{15 x^{2}}{6 a^{2}} \)

By doing this, we've simplified the division problem into a manageable multiplication exercise, paving the way for the next step: simplification.

Simplification of Algebraic Expressions

Simplifying algebraic expressions helps in reducing complex equations into forms that are easier to understand and use. After multiplying the fractions in our exercise, we obtain a large expression that needs simplification:\( \frac{12a^2b^3 \cdot 15x^2}{-5xy^4 \cdot 6a^2} \).

• First, identify common factors in the numerator and denominator, such as \(a^2\) and \(6\).
• Cancel out the common terms that appear in both the numerator and denominator.
• Divide through by these common factors to simplify further.

This results in the simplified expression: \( \frac{-30 b^3 x^2}{y^4} \). Simplification not only makes calculations easier but also provides clearer insights into the behavior and character of the mathematical expression.