Problem 13 - Algebra 2

Question

Divide. State any restrictions on the variables. $$ \frac{7 x}{4 y^{3}} \div \frac{21 x^{3}}{8 y} $$

Step-by-Step Solution

Answer

\(\frac{x^{2}}{2y^{2}}\) where \(y != 0\)

Step 1: Simplify the Dividend and Divisor

Rewrite the expression \(\frac{7 x}{4 y^{3}} \div \frac{21 x^{3}}{8 y}\) as \(\frac{7 x}{4 y^{3}} * \frac{8 y}{21 x^{3}}\) . It's important to remember that dividing by a fraction is equivalent to multiplying by its reciprocal.

Step 2: Simplify the Numerator and Denominator

Simplify the expression by cancelling out common terms in the numerator and denominator. The result is \(\frac{(7/21)x/x^{3}}{(4/8)y^{3}/y}\), which simplifies to \(\frac{x^{2}}{2y^{2}}\). This was achieved by cancelling out 7 with 21 to be 1 and 3 in the numerator, x with x^3 to result in x^2, 4 with 8 to be 1 and 2 in the denominator and y^3 with y to result in y^2.

Step 3: State the Restrictions

With an expression that contains variables in the denominator, we need to state the restrictions on the variables, as division by zero is undefined. The denominator is \(2y^{2}\), so the restriction is that \(y != 0\).

Key Concepts

Understanding Fractions in Division

Fractions are an essential part of algebraic division, especially when dealing with variables. In essence, a fraction consists of a numerator, the top part, and a denominator, the bottom part. When dividing by a fraction, it's crucial to know that this is the same as multiplying by its reciprocal. The reciprocal of a fraction is simply switching its numerator and denominator. By converting division into multiplication, you can more easily simplify and resolve the equation. Pay attention to how each component interacts in the process, particularly in complex fractions with variables involved.

Manipulating Variables Effectively

Variables are symbols that are used to represent unspecified numbers or values; in this example, they are denoted by 'x' and 'y.' When working with variables in a division problem, you'll handle them just like any number. However, specific attention should be given to their exponents during simplification. Simplification often involves cancelling out terms; for instance, dividing like variables require subtracting their exponents. Notably, in our problem, the term \(x\) in the numerator and \(x^3\) in the denominator result in \(x^2\) when simplified. This operation is key to ensuring that you reduce the expression to its simplest form.

The Simplification Process

Simplification is all about making expressions as concise as possible. In algebra, this means combining like terms and reducing expressions through operations such as cancelling. When simplifying \(\frac{7x}{4y^3} \cdot \frac{8y}{21x^3}\), notice

• \(7\) and \(21\) reduce to \(1\) and \(3\).
• \(x\) cancels out one \(x\) from \(x^3\) resulting in \(x^2\).
• \(4\) and \(8\) reduce to \(1\) and \(2\).
• \(y^3\) cancels out one \(y\) from \(y\) yielding \(y^2\).

This leads to a simpler expression of \(\frac{x^2}{2y^2}\), which is often easier to interpret and use in subsequent calculations or problem-solving steps.

Navigating Restrictions in Expressions

When dealing with fractions involving variables, restrictions are necessary conditions that must be stated to avoid undefined expressions. This is mainly owed to the fact that division by zero is not permissible in mathematics. For the expression or solution to remain valid, ensure no variable results in a zero denominator. In the expression \(\frac{x^2}{2y^2}\), the denominator contains \(y^2\). Therefore, \(y\) must not be equal to zero. Recognizing and stating these restrictions is crucial for avoiding mathematical errors or contradictions in your solution.