Problem 69 - Prealgebra and Introductory Algebra

Question

Divide. $$\frac{x^{2} y^{5}}{x^{2}-11 x+30} \div \frac{x y^{6}}{x^{2}-7 x+10}$$

Step-by-Step Solution

Answer

The simplified form of the given expression is \( \frac{(x-2)}{y(x-6)} \).

Step 1: Rewrite the division as multiplication

Any division can be rewritten as a multiplication problem. Division by a fraction is the same as multiplication by its reciprocal. Rewrite the given expression as \[\frac{x^{2} y^{5}}{x^{2}-11 x+30} \times \frac{x^{2}-7 x+10}{x y^{6}}\]

Step 2: Simplify numerators and denominators separately

Look if anything could be factored out or simplified in each of the four parts of the expression, which are two numerators and two denominators. There are no common factors to simplify in numerators. But in denominators, both of them are quadratic expressions and they can be factored. We get\[\frac{x^{2} y^{5}}{(x-5)(x-6)} \times \frac{(x-5)(x-2)}{x y^{6}}\]

Step 3: Cancel common factors between numerators and denominators

After factoring, now look for common factors between numerators and denominators and cancel them out. \( x^2 \) is common in numerator and denominator, \( y^5 \) is common, and \( x-5 \) is common. Cancel these common factors out. We obtain\[\frac{x y^{5}}{(x-5)(x-6)} \times \frac{(x-5)(x-2)}{x y^{6}} = \frac{1}{x-6} \times \frac{x-2}{ y}\]

Step 4: Multiply the final fractions

As a final step, multiply the numerators together and denominators together of these two fractions. We get final answer as \[\frac{(x-2)}{y(x-6)} \]

Key Concepts

Multiplication of Fractions

When multiplying fractions, a simple rule applies: multiply the numerators together and the denominators together. This process is consistent whether you’re dealing with numbers or variables. In our given exercise, the key step is to transition the division into multiplication by using the reciprocal.

• First, rewrite the division problem. Turn the second rational expression upside-down to use its reciprocal. This changes the division into multiplication.
• Next, proceed to multiply the numerators across and the denominators across. This step combines both fractions into one.

For example, \[\frac{a}{b} \times \frac{c}{d} = \frac{a \cdot c}{b \cdot d}\]is a representation of multiplying two simple fractions. Remember, simplifying fractions beforehand can greatly ease this process.

Factoring Quadratic Expressions

When working with rational expressions, especially those involving quadratics, factoring is an essential skill. Quadratic expressions often appear as denominators within rational expressions, and simplifying them is a must.

• Look for expressions of the form \(ax^2 + bx + c\), which can often be factored into two binomials, such as \((x-m)(x-n)\).
• It's essential to identify numbers or expressions that multiply to give \(c\) and add to give \(b\). This skill is honed through practice.

In the original exercise, the expressions \(x^2 - 11x + 30\) and \(x^2 - 7x + 10\) are factored to allow for easier simplification. Factoring them gives us:\[x^2 - 11x + 30 = (x-5)(x-6)\]\[x^2 - 7x + 10 = (x-5)(x-2)\]Factoring quadratics can sometimes reveal common factors, further simplifying the expression.

Simplifying Algebraic Expressions

Simplifying algebraic expressions is akin to tidying up; you clear out unnecessary or repetitive elements, leaving a straightforward, concise expression. In rational expressions, this often involves canceling common factors.

• After factoring, compare the numerators and denominators for common terms or factors that can be canceled out.
• Simplification reduces complexity and can make manual calculations easier to manage.
• Alignment of terms and cancellation of similar expressions is crucial in simplifying without losing the essence of the original expression.

In our exercise, we canceled common factors such as \(x\), \(y^5\), and \(x-5\). This process reduced the expression significantly:\[\frac{x^2 y^5}{(x-5)(x-6)} \times \frac{(x-5)(x-2)}{x y^6}\]to\[\frac{1}{x-6} \times \frac{x-2}{y} \Rightarrow \frac{x-2}{y(x-6)}\]This final simplification not only makes the expression more manageable but also ensures that it represents the same mathematical quantity as the original complex form.