Problem 11 - Algebra 2

Question

Simplify each expression. \(\frac{20 x y^{3}}{21} \div \frac{15 x^{3} y^{2}}{14}\)

Step-by-Step Solution

Answer

The simplified expression is \( \frac{8y}{9x^2} \).

Step 1: Understand the Division of Fractions

To divide fractions, we multiply by the reciprocal of the divisor. For the given expression \( \frac{20xy^3}{21} \div \frac{15x^3 y^2}{14} \), we need to multiply \( \frac{20xy^3}{21} \) by the reciprocal of \( \frac{15x^3 y^2}{14} \), which is \( \frac{14}{15x^3 y^2} \).

Step 2: Set Up the Multiplication

Now, set up the multiplication of the fractions: \( \frac{20xy^3}{21} \times \frac{14}{15x^3 y^2} \).

Step 3: Multiply the Fractions

Multiply the numerators: \(20xy^3 \times 14 = 280xy^3 \). Multiply the denominators: \(21 \times 15x^3 y^2 = 315x^3 y^2 \). So, the expression becomes \( \frac{280xy^3}{315x^3 y^2} \).

Step 4: Simplify the Expression

Start by simplifying the coefficients: \( \frac{280}{315} \). Both 280 and 315 can be divided by 35, leading to \( \frac{8}{9} \). Next, simplify the variables: \( \frac{xy^3}{x^3 y^2} = \frac{x^{1-3} y^{3-2}}{1} = \frac{y}{x^2} \). Thus, the final simplified expression is \( \frac{8y}{9x^2} \).

Key Concepts

Simplifying Expressions

Simplifying expressions involves reducing them to their simplest form. In our exercise, we simplify an algebraic expression derived from dividing two fractions.
This process consists of working systematically to make the expression less complicated.

• Start with understanding the components, such as numbers and variables' powers.
• Identify common factors in both the numerator and denominator.
• Cancel these common factors to simplify.

By breaking down complex fractions into easier parts, you gain clearer insight into the relationship between numbers and variables. This simplification is what ultimately gives us the neatest representation of an expression.

Division of Fractions

Dividing fractions sounds more complicated than it is. A key trick is knowing that dividing by a fraction is the same as multiplying by its reciprocal.
When faced with a fraction division like in our exercise, you should:

• Identify the divisor, which is the fraction you are dividing by.
• Find the reciprocal of this divisor. The reciprocal of a fraction is created by swapping its numerator and denominator.
• Change the division operation into a multiplication with the reciprocal.

In our example, we divide two algebraic fractions, transforming the process into a multiplication step by using reciprocity. Understanding this concept is essential for handling more complex fraction divisions efficiently.

Reciprocal

The concept of a reciprocal is quite handy, especially in division of fractions. Simply put, the reciprocal of a number is what you multiply by to get one.
For any fraction, its reciprocal is just flipping its numerator and denominator. For instance, the reciprocal of \(\frac{a}{b}\) is \(\frac{b}{a}\).

• This is crucial when dividing by a fraction because multiplying by its reciprocal is always necessary.
• This property ensures that division can be shifted to multiplication, simplifying calculations.
• Using reciprocal effectively turns complex fraction operations into simpler multiplication tasks.

Reciprocal plays a central role in algebraic fraction manipulation, offering a straightforward path to simplifying otherwise tricky problems.

Multiplication of Fractions

Multiplying fractions is straightforward once the idea of reciprocals is clear. You simply multiply the numerators together, and then the denominators together.
Here's how you do it:

• Write down both fractions as they appear in the multiplication step.
• Multiply the top numbers (numerators) to get the new numerator.
• Multiply the bottom numbers (denominators) to get the new denominator.

In this exercise, multiplying \(\frac{20xy^3}{21}\) by the reciprocal \(\frac{14}{15x^3 y^2}\) illustrated this. After you multiply, the result often needs further simplification, reducing it to its simplest form, thus finishing the problem neatly. Understanding multiplication of fractions is a major key in managing complex algebraic fractions.