Problem 42 - Elementary Algebra

Question

Perform the indicated operations and express the answers in simplest form. Remember that multiplications and divisions are done in the order that they appear from left to right. $$\frac{5 x y^{2}}{12 y} \cdot \frac{18 x^{2}}{15 y} \div \frac{3}{2 x y}$$

Step-by-Step Solution

Answer

\( \frac{x^4y}{3} \)

Step 1: Simplify the first multiplication

We start with the expression \( \frac{5xy^2}{12y} \cdot \frac{18x^2}{15y} \). To simplify, multiply the numerators and the denominators separately.Numerator: \( 5xy^2 \times 18x^2 = 90x^3y^2 \).Denominator: \( 12y \times 15y = 180y^2 \).Now, the fraction is \( \frac{90x^3y^2}{180y^2} \).

Step 2: Simplify the fraction

We divide both the numerator and the denominator by 90 to simplify the fraction. We also cancel \( y^2 \) from both the numerator and the denominator:\[ \frac{90x^3y^2}{180y^2} = \frac{x^3}{2} \].

Step 3: Simplify the Division

Now, we divide the simplified product from Step 2 by the third fraction \( \frac{3}{2xy} \).Performing the division by a fraction is equivalent to multiplying by its reciprocal:\[ \frac{x^3}{2} \div \frac{3}{2xy} = \frac{x^3}{2} \times \frac{2xy}{3} \].

Step 4: Complete the multiplication

Now, multiply \( \frac{x^3}{2} \) by \( \frac{2xy}{3} \):Numerator: \( x^3 \times 2xy = 2x^4y \).Denominator: \( 2 \times 3 = 6 \).So, the fraction becomes \( \frac{2x^4y}{6} \).

Step 5: Simplify the result

Simplify \( \frac{2x^4y}{6} \) by dividing both the numerator and the denominator by 2:\[ \frac{x^4y}{3} \].

Key Concepts

Simplifying Expressions

Simplifying expressions is a crucial skill in algebra, as it makes complex equations easier to work with. To simplify an expression, we need to combine like terms and reduce fractions whenever possible. For example, when given the expression \( \frac{5xy^2}{12y} \cdot \frac{18x^2}{15y} \), the goal is to make it as simple as possible. This involves working with both the numerators and denominators.

First, we multiply the numerators: \( 5xy^2 \times 18x^2 = 90x^3y^2 \). Next, we do the same with the denominators: \( 12y \times 15y = 180y^2 \).

• Reduce fractions by dividing the numerator and denominator by their greatest common factor.
• In the expression \( \frac{90x^3y^2}{180y^2} \), divide by 90 and cancel \( y^2 \), resulting in \( \frac{x^3}{2} \).

Simplifying expressions helps manage the complexity of algebraic operations.

Multiplication of Fractions

Multiplying fractions is systematic and straightforward. You simply multiply the numerators with each other and the denominators with each other. This holds true for both algebraic and numerical fractions.

For instance, in \( \frac{5xy^2}{12y} \cdot \frac{18x^2}{15y} \), we multiply:

• Numerators: \( 5xy^2 \times 18x^2 = 90x^3y^2 \)
• Denominators: \( 12y \times 15y = 180y^2 \)

This process helps transform multiple fractions into a single one, easing the simplification process later. Always remember to simplify the resulting fraction.

Division of Fractions

Dividing fractions involves a simple trick that turns the division into multiplication, making it easier. You flip (or find the reciprocal of) the second fraction and multiply. This technique can dramatically simplify the work involved.

For the expression \( \frac{x^3}{2} \div \frac{3}{2xy} \), transform it by multiplying by the reciprocal of \( \frac{3}{2xy} \), which is \( \frac{2xy}{3} \). So, it becomes:

• \( \frac{x^3}{2} \times \frac{2xy}{3} \)
• The operation is now multiplication, avoiding direct division, which often results in errors.

Ensure, as always, to simplify the resulting fraction to remain manageable.

Canceling Common Factors

Canceling common factors is a technique used to simplify fractions further. It involves finding and eliminating the same variables or numbers in both the numerator and the denominator. This reveals the simplest form of the fraction.

After some calculations, you might end with \( \frac{2x^4y}{6} \). Look for common factors in both parts of the fraction. Here, the number 2 is common:

• Divide both the numerator and denominator by the common factor: \( \frac{x^4y}{3} \).

By canceling, we ensure our expression is concise. Reducing fractions to their simplest form not only helps with clarity but also with any further operations needed.