Problem 7 - Elementary and Intermediate Algebra

Question

Fill in the blanks. $$ \frac{\frac{12}{y^{2}}}{\frac{4}{y^{3}}} \text { means } \frac{12}{y^{2}} \quad \frac{4}{y^{3}} $$

Step-by-Step Solution

Answer

3y

Step 1: Understand the original expression

The expression given is \( \frac{\frac{12}{y^2}}{\frac{4}{y^3}} \). This can be understood as dividing one fraction by another fraction.

Step 2: Simplify the division of fractions

To divide fractions, you multiply by the reciprocal of the divisor. Therefore, \( \frac{\frac{12}{y^2}}{\frac{4}{y^3}} \) becomes \( \frac{12}{y^2} \times \frac{y^3}{4} \).

Step 3: Multiply the numerators and denominators

Now, perform the multiplication across the numerators and the denominators. This gives:\[\frac{12 \times y^3}{y^2 \times 4}.\]

Step 4: Simplify the expression

Simplify the expression by cancelling common terms. The 12 and the 4 have a common factor, which is 4:\[\frac{12y^3}{4y^2} = \frac{12}{4} \times \frac{y^3}{y^2} = 3 \times y = 3y.\]

Step 5: Final Step: Write the simplified expression

The simplified expression of the given division is \( 3y \). Therefore, the expression \( \frac{12}{y^2} \) divided by \( \frac{4}{y^3} \) simplifies to \( 3y \).

Key Concepts

Simplifying Fractions

Simplifying fractions involves reducing them to their simplest form. This means ensuring that both the numerator and the denominator have no common divisors other than 1.
To simplify fractions:

• Find the greatest common divisor (GCD) of the numerator and the denominator.
• Divide both the numerator and the denominator by the GCD.

This reduces the fraction to its simplest form, making further calculations easier.
For example, in the solution, after multiplying, the expression is \( \frac{12y^3}{4y^2} \). Here, 12 and 4 share a common factor, which is 4. Simplifying gives \( \frac{12}{4} \), resulting in 3, which is part of the final simplified expression.

Division of Fractions

Dividing fractions might seem complicated at first, but it's more about flipping and multiplying. Instead of division, you change the division into a multiplication by using the reciprocal.

• The reciprocal is made by swapping the numerator and the denominator of the fraction you are dividing by.
• Multiplication is then done by multiplying the numerators and the denominators separately.

For instance, dividing \( \frac{12}{y^2} \) by \( \frac{4}{y^3} \) converts to multiplication as \( \frac{12}{y^2} \times \frac{y^3}{4} \). This technique simplifies the process and helps in easily handling operations with fractions.

Reciprocals

Understanding reciprocals are crucial in dividing fractions. The reciprocal of a fraction changes the position of the numerator and the denominator.
For fraction \( \frac{a}{b} \), the reciprocal is \( \frac{b}{a} \).
Using reciprocals:

• Helps in turning a division operation into a multiplication one.
• Is essential for simplifying the expression correctly.

In our exercise, \( \frac{4}{y^3} \) is turned into its reciprocal \( \frac{y^3}{4} \) for the multiplication step. Without understanding and using reciprocals, the division of fractions would remain unnecessarily complicated.

Polynomial Expressions

Polynomial expressions involve variables raised to whole number powers and combined using addition, subtraction, and multiplication.
When working with polynomial expressions in fractions, simplification often requires dividing through by common terms or multiplying terms.
This usually involves knowing how to:

• Identify like terms.
• Use distributive laws.
• Factor out common variables.
• Simplify cross-multiplications.

In the exercise, after applying reciprocals and multiplying \( \frac{12}{y^2} \times \frac{y^3}{4} \), the expression simplifies to \( 3y \). Recognizing that \( y^3 \) and \( y^2 \) share \( y^2 \) as a common factor, helps simplify and determine the final result \( 3y \). Understanding these concepts is key to effectively solving polynomial fractions.