Problem 6 - Algebra A Combined Function

Question

Find each product and simplify if possible. See Examples 1 through 3. $$ -\frac{9 x^{3} y^{2}}{18 x y^{5}} \cdot y^{3} $$

Step-by-Step Solution

Answer

The simplified product is \(-\frac{x^2}{2}\).

Step 1: Rewrite the Expression

The given expression is \(-\frac{9x^3y^2}{18xy^5} \cdot y^3\). First, rewrite it in a single fraction for clarity: \(-\frac{9x^3y^2 \cdot y^3}{18xy^5}\).

Step 2: Simplify the Coefficients

Look at the numerical coefficients, \(-9\) and \(18\). Simplify them as a fraction: \(-\frac{9}{18} = -\frac{1}{2}\).

Step 3: Simplify the Variable \(x\)

Now simplify the expression for \(x\). We have \(x^3\) in the numerator and \(x\) in the denominator: \(\frac{x^3}{x} = x^{3-1} = x^2\).

Step 4: Simplify the Variable \(y\)

For the \(y\) terms, we have \(y^2 \cdot y^3 = y^{2+3} = y^5\) in the numerator and \(y^5\) in the denominator. So, \(\frac{y^5}{y^5} = y^0 = 1\).

Step 5: Combine and Simplify

Combine the simplified parts from the previous steps: \(-\frac{1}{2} \cdot x^2 \cdot 1\). This simplifies to \(-\frac{x^2}{2}\).

Step 6: Final Result

The simplified product of the expression is \(-\frac{x^2}{2}\).

Key Concepts

Understanding Rational Expressions

Rational expressions might sound complicated, but they are similar to fractions, except that they include variables. A rational expression is essentially any expression you can write as a fraction or ratio of two polynomials. For example, \(-\frac{9x^3y^2}{18xy^5}\) is a rational expression because it involves the variables \(x\) and \(y\) in polynomial form, divided by another polynomial.
When working with rational expressions, the rules you use for fractions still apply. This means

• Finding a common denominator when adding or subtracting.
• Multiplying directly across the numerator and denominator.
• Simplifying by canceling common factors in the numerator and the denominator, just as you do with fractions.

Understanding rational expressions is just taking your fraction skills to a new level by including algebraic elements.

Rules for Multiplying Fractions

Multiplying fractions is quite straightforward. Just like with regular numbers, when you multiply fractions, you multiply the numerators together and the denominators together.
Given the rational expression \(\left(-\frac{9x^3y^2}{18xy^5}\right) \cdot y^3\), you can rewrite it as a single fraction in order to see the multiplication clearly:\(-\frac{9x^3y^2 \, \cdot \, y^3}{18xy^5}\).Here are the steps for multiplication:

• Multiply the numerators together: This involves combining the coefficients and the variables in the numerator.
• Multiply the denominators together: Because in this case the denominator remains 18xy^5, you only combine like terms.
• Simplify the result: As with any fraction, you must look for and cancel out common factors.

By consistently applying these steps, working with multiplying fractions becomes a matter of routine.

Simplifying Exponents Made Easy

When you're dealing with exponents, there are some handy rules to keep in mind that make simplifying expressions a breeze. One such rule is \(a^{m+n} = a^m \, \cdot \, a^n\). This came in useful when simplifying the expression \(y^2 \cdot y^3\) to produce \(y^5\). Additionally, whenever you have the same base present in both the numerator and denominator, you can subtract the exponents: \(\frac{a^m}{a^n} = a^{m-n}\). This rule helped us to cancel out \(y^5\) from both parts to get \(y^0\), which equals 1.
Keep in mind these key rules for exponents:

• Multiply powers with the same base: Add their exponents.
• Divide powers with the same base: Subtract their exponents.
• Any non-zero number raised to the power zero equals 1.

These simple rules allow you to handle any expression involving exponents with confidence and simplicity.