Problem 49 - Elementary Algebra

Question

For exercises 39-82, simplify. $$ \frac{9 x^{2}}{8 y} \div \frac{x}{y} $$

Step-by-Step Solution

Answer

\frac{9x}{8}

Step 1 - Rewrite the Division

Rewrite the division of fractions as a multiplication by the reciprocal. The division \(\frac{9 x^{2}}{8 y} \div \frac{x}{y}\) becomes \(\frac{9 x^{2}}{8 y} \times \frac{y}{x}\).

Step 2 - Multiply the Numerators and Denominators

Multiply the numerators together and the denominators together: \(\frac{9 x^{2} \times y}{8 y \times x}\).

Step 3 - Simplify the Expression

Combine the terms in both the numerator and the denominator: \(\frac{9 x^{2} y}{8 y x}\).

Step 4 - Cancel Common Factors

Cancel out the common terms in the numerator and the denominator. The \(y\) term cancels out, leaving: \(\frac{9 x^{2}}{8 x} \). Then cancel one \(x\) term from the numerator and denominator resulting in \(\frac{9 x}{8}\).

Key Concepts

Algebraic Manipulation

Algebraic manipulation involves rearranging and simplifying equations and expressions to make them easier to work with. One common tactic is to transform division into multiplication by using the reciprocal of the divisor. For example, when you have a problem like \(\frac{9 x^{2}}{8 y} \div \frac{x}{y}\), you can rewrite it as \(\frac{9 x^{2}}{8 y} \times \frac{y}{x}\). This makes it simpler to manage and sets up the problem for the next steps. Always rewrite division problems this way to pave the way for easier multiplication and further simplification.

Fraction Division

Fraction division is made easier by converting to multiplication. When you divide by a fraction, like \(\frac{9 x^{2}}{8 y} \div \frac{x}{y}\), it's equivalent to multiplying by its reciprocal. The reciprocal of \(\frac{x}{y}\) is \(\frac{y}{x}\). So the rewritten problem becomes \(\frac{9 x^{2}}{8 y} \times \frac{y}{x}\).
This step is crucial because multiplying fractions is much simpler and it aligns with familiar multiplication rules, allowing you to streamline the problem. In effect, changing division into multiplication simplifies the operation and sets you up for easier numerical or algebraic manipulation.

Simplifying Expressions

Simplifying expressions means breaking them down to their simplest form. After rewriting our problem from division to multiplication, you'll multiply the numerators together and the denominators together. So, \(\frac{9 x^{2}}{8 y} \times \frac{y}{x}\) becomes \(\frac{9 x^{2} \times y}{8 y \times x}\).
Next comes combining terms. Multiply the variables and constants in the numerator and do likewise for the denominator. This results in \(\frac{9 x^{2} y}{8 y x}\). Simplified expressions are easier to understand and work with, and this step ensures everything is in the simplest, most manageable form.

Canceling Common Factors

The final step in simplifying algebraic fractions is canceling out common factors. In the expression \(\frac{9 x^{2} y}{8 y x}\), notice the \( y \) in the numerator and denominator cancels out, leaving \(\frac{9 x^{2}}{8 x} \). Then, cancel one \( x \) from the numerator and denominator, leading to \(\frac{9 x}{8} \).
Canceling common factors reduces the expression to its simplest form. It's a critical step because it removes redundant terms, making the result cleaner and easier to understand. Always look for common factors between the numerator and denominator to simplify your expression to the fullest.