Problem 64 - Algebra 2

Question

Divide. State any restrictions on the variables. \(\frac{6 x^{2}}{y} \div \frac{12 x^{4}}{y^{3}}\)

Step-by-Step Solution

Answer

The result of dividing \(\frac{6 x^{2}}{y} \div \frac{12 x^{4}}{y^{3}}\) is \(\frac{y^{2}}{2x^{2}}\) under the restrictions \(x ≠ 0\) and \(y ≠ 0\).

Step 1: Rewrite the division as multiplication

Rewrite \(\frac{6 x^{2}}{y} \div \frac{12 x^{4}}{y^{3}}\) as \(\frac{6 x^{2}}{y} * \frac{y^{3}}{12 x^{4}}\). Notice that dividing by \(\frac{12 x^{4}}{y^{3}}\) is the same as multiplying by its reciprocal, \(\frac{y^{3}}{12 x^{4}}\).

Step 2: Multiply the fractions

Now, multiply the fractions together: \((\frac{6 x^{2}}{y}) * (\frac{y^{3}}{12 x^{4}}) = \frac{6x^{2} * y^{3}}{y * 12x^{4}}\) = \(\frac{6y^{2}}{12x^{2}}\)

Step 3: Simplify the result

\(\frac{6y^{2}}{12x^{2}}\) simplifies to \(\frac{1}{2}* \frac{y^{2}}{x^{2}}\) or \(\frac{y^{2}}{2x^{2}}\) when reducing the fractions. So this is the result of dividing the two given rational expressions.

Step 4: Determine restrictions on the variables

At no point should the denominator of any fraction get to be zero as it would make the expression undefined. Hence, \(y\) and \(x\) must not be zero. So the restrictions are \(x ≠ 0\) and \(y ≠ 0\).

Key Concepts

Simplifying Rational Expressions

Simplifying rational expressions is like simplifying fractions, but they can involve polynomials instead of just numbers. Let's break down the process:

• First, identify the numerator and the denominator in a rational expression. They can be polynomials or simple terms just like numbers.
• Common factors between the numerator and denominator should be canceled out, just as you would with a regular fraction. This often involves factoring to identify common elements.

In the given exercise, you begin with the expressions \(\frac{6 x^{2}}{y}\) and \(\frac{12 x^{4}}{y^{3}}\). After rewriting the problem as a multiplication with a reciprocal, you end up simplifying \(\frac{6x^{2}y^{3}}{12x^{4}y}\).

By canceling out common factors like \(x^{2}\) from the numerator and denominator, as well as simplifying the coefficients 6 and 12, you reach the expression \(\frac{y^{2}}{2x^{2}}\).

Simplification makes it easier to understand expressions and solve problems without unnecessary complexity.

Restrictions on Variables

Recognizing restrictions on variables is crucial when dividing rational expressions. These restrictions arise because a denominator cannot be zero — it makes the expression undefined.

Here’s how to find and interpret these restrictions:

• Examine each term in the denominator across all steps of your calculation. Set these expressions not equal to zero (\(eq 0\)).
• Watch all calculations that occur during simplification, as cancellation of terms can impact variables present in denominators.

For the task at hand, notice that in the original and simplified expressions, the denominators contain \(y\) and \(x\). Therefore, both \(x\) and \(y\) must be nonzero. Hence, the restrictions are \(x eq 0\) and \(y eq 0\).

Understanding these restrictions helps avoid undefined operations and keeps calculations rooted in mathematically sound principles.

Multiplication of Fractions

The multiplication of fractions, whether they're simple numbers or complex rational expressions, follows a straightforward process:

• Multiply the numerators together to get the new numerator.
• Multiply the denominators together to get the new denominator.

In rational expressions, you must handle variables alongside numbers during multiplication. In our initial expression, after rewriting the division as multiplication, we performed the operation: \((\frac{6 x^{2}}{y}) \times (\frac{y^{3}}{12 x^{4}})\). This transformed into \(\frac{6x^{2}y^{3}}{12x^{4}y}\).

Remember to keep track of coefficients and similar terms, and simplify wherever possible. The simplification of these multiplied factors brought us to \(\frac{y^{2}}{2x^{2}}\).

Multiplying fractions consolidates steps and can make other processes, like simplification and identifying restrictions, significantly easier.