Problem 16 - Elementary Algebra

Question

Perform the indicated multiplications and divisions and express your answers in simplest form. $$14 x y^{2} \div \frac{7 y}{9}$$

Step-by-Step Solution

Answer

The simplified expression is \(18xy\).

Step 1: Identify the given expression

The original expression given is \(14xy^2 \div \frac{7y}{9}\). It involves dividing the product \(14xy^2\) by the fraction \(\frac{7y}{9}\).

Step 2: Change division to multiplication

When dividing by a fraction, you can multiply by its reciprocal. Therefore, the expression becomes \[ 14xy^2 \times \frac{9}{7y} \].

Step 3: Multiply expressions

Multiply the fractions by multiplying the numerators and denominators. The expression becomes:\[ \frac{14xy^2 \times 9}{7y} = \frac{126xy^2}{7y} \].

Step 4: Simplify the expression

Divide the numerator and the denominator by any common factors. In this case, both \(126\) and \(7\) have a common factor of \(7\), and \(y^2\) divided by \(y\) simplifies to \(y\). Thus, the expression simplifies to:\[ \frac{126 \div 7}{7y \div 7} \times \frac{xy^2}{y} = 18xy \].

Key Concepts

Multiplication and Division of Fractions

When exploring the multiplication and division of fractions, it's crucial to understand how these operations function. Let's start with multiplication. Multiplying fractions is quite straightforward. You simply multiply the numerators together and then the denominators together.
For example:

• For \( \frac{a}{b} \times \frac{c}{d} \), the result would be \( \frac{ac}{bd} \).

On the other hand, division of fractions might seem a bit more complex but follows a simple rule: when you divide by a fraction, you multiply by its reciprocal.

• The reciprocal of a fraction \( \frac{a}{b} \) is \( \frac{b}{a} \).

This means whenever you encounter a division, like \( \frac{m}{n} \div \frac{p}{q} \), you change it to multiplication: \( \frac{m}{n} \times \frac{q}{p} \). By doing so, you apply the same multiplication rule to solve the problem.

Simplifying Algebraic Expressions

Simplifying algebraic expressions is a crucial skill in algebra. It involves combining like terms and reducing expressions to their simplest form for easier interpretation and calculation. Let's explore how this works.
The process generally includes:

• Combining like terms: Terms that contain the same variables raised to the same power.
• Reducing fractions: Dividing the numerator and the denominator by their greatest common factor (GCF).

In our exercise, we saw the expression \( \frac{126xy^2}{7y} \). To simplify this, we divided both the numerator and the denominator by the common factor \( 7 \), which yields \( \frac{18xy^2}{1y} \). Additionally, \(y^2\) divided by \(y\) simplifies to \(y\), leading to \(18xy\).
This makes the expression both simpler and more accessible for further mathematical operations.

Reciprocal of a Fraction

Understanding the reciprocal of a fraction is key to dividing fractions easily. The reciprocal is simply what you flip upside down. If you have a fraction \( \frac{a}{b} \), its reciprocal would be \( \frac{b}{a} \).
Reciprocals play a significant role when dividing fractions because they allow us to convert division operations into multiplication ones, which are typically easier to handle.For instance:

• If you encounter \( x \div \frac{7y}{9} \), it becomes \( x \times \frac{9}{7y} \) after taking the reciprocal of \( \frac{7y}{9} \).

By converting to multiplication using reciprocals, the calculations involve straightforward multiplication of numerators and denominators, simplifying complex algebraic manipulation into more manageable steps.