Problem 7 - Basic Technical Mathematics with Calculus

Question

Simplify the given expressions involving the indicated multiplications and divisions. $$\frac{4 x}{3 y} \times \frac{9 y^{2}}{2}$$

Step-by-Step Solution

Answer

The simplified expression is \(6xy\).

Step 1: Rewrite the Expression

Start by rewriting the expression to visually separate the numerators and denominators. The expression we're simplifying is: \( \frac{4x}{3y} \times \frac{9y^2}{2} \). This is the same as \( \frac{4x \cdot 9y^2}{3y \cdot 2} \).

Step 2: Multiply Across the Numerators and Denominators

Now, multiply the numerators together and the denominators together. Numerator: \( 4x \cdot 9y^2 = 36xy^2 \). Denominator: \( 3y \cdot 2 = 6y \). This gives us the new expression: \( \frac{36xy^2}{6y} \).

Step 3: Simplify the Fraction

To simplify \( \frac{36xy^2}{6y} \), divide both the numerator and the denominator by the common factor. Both 36 and 6 can be divided by 6. Simplified, the fraction becomes \( \frac{6xy^2}{y} \).

Step 4: Simplify by Cancelling Terms

Further simplify \( \frac{6xy^2}{y} \) by cancelling out the common \( y \) term in the numerator and denominator. This leaves \( 6xy^{2-1} = 6xy \).

Key Concepts

Multiplication and Division of Fractions

When multiplying fractions, it's essential to understand that you simply multiply the numerators together and the denominators together. This makes it straightforward and manageable, as you don't have to find a common denominator like you would when adding or subtracting fractions.

In this context, multiplying fractions involves these steps:

• Multiply the numerators: Take the top number from both fractions and multiply them.
• Multiply the denominators: Take the bottom number from both fractions and multiply them.

This gives you a new fraction with a multiplied numerator and denominator. In our original exercise, we saw this process in action:

• Numerator: \(4x \times 9y^2 = 36xy^2\)
• Denominator: \(3y \times 2 = 6y\)

However, unlike multiplication, when dealing with division of fractions, you have to remember to multiply by the reciprocal. When dividing two fractions, instead of dividing directly, you multiply the first fraction by the reciprocal of the second. Simplifying fractions in this step tends to make the next steps much easier.

Numerators and Denominators

In fractions, the numerator and denominator play a critical role. The numerator is the top number, indicating the parts we have, while the denominator, the bottom number, signifies the parts required to make up a whole.

Understanding these components is crucial for performing operations on fractions:

• In multiplication, both numerators directly influence the new numerator.
• Similarly, the denominators determine the new denominator.
• When simplifying, you're often looking for common factors shared between the numerator and denominator.

In our case, each fraction's components must be multiplied to get a new fraction. For example:

• Original numerators are \(4x\) and \(9y^2\), resulting in the new numerator \(36xy^2\).
• Original denominators are \(3y\) and \(2\), which combine to a new denominator \(6y\).

Simplifying often involves dividing each by the greatest common factor, making the fraction easier to grasp and ensuring its simplest form.

Simplifying Fractions

Simplifying fractions involves reducing them to their simplest form so they are easier to work with. This means making the numerator and denominator as small as possible without changing the overall value of the fraction.

The process of simplification includes:

• Identifying common factors that both the numerator and denominator share.
• Dividing both parts by these common factors to reduce the fraction.

In our example, we start with \(\frac{36xy^2}{6y}\). Notice both 36 and 6 can be divided by 6, simplifying it to \(\frac{6xy^2}{y}\).

Next, we recognize that both the numerator and denominator have a "\(y\)" term. By canceling these terms, we simplify further to \(6xy^{2-1} = 6xy\). This final expression is in its simplest form, stripped of all common factors, making it straightforward to understand and use.