Problem 338 - Intermediate Algebra

Question

In the following exercises, simplify. $$ 0 \div\left(y-\frac{1}{6}\right), \text { where } x \neq \frac{1}{6} $$

Step-by-Step Solution

Answer

The expression simplifies to 0.

Step 1: Understand the problem

The goal of this exercise is to simplify the expression \(0 \div \left(y - \frac{1}{6}\right)\) where \( y \eq \frac{1}{6} \).

Step 2: Apply the division rule

Recall that any number divided by another number (except zero) will be zero. Therefore, \(0 \div (y - \frac{1}{6}) \) simplifies directly to 0, regardless of any value of \( y \) provided \( y \eq \frac{1}{6} \).

Step 3: State the simplified form

Since \( y \eq \frac{1}{6}\), \( 0 \div \left(y - \frac{1}{6}\right) \) simplifies to 0.

Key Concepts

division rules

Understanding division rules is key to mastering algebra. One of the simplest yet most important rules to remember is that any number divided by another number (except zero) will result in zero. For example, the expression \( 0 \div \left(y - \frac{1}{6}\right) \) simplifies to zero. This is because division by any non-zero number cannot change the fact that the numerator (the number being divided) is zero. Always take note that if you're dividing by zero, the result is undefined, and you cannot perform the division.

rational expressions

Rational expressions are simply fractions where both the numerator and the denominator are polynomials. In our example, we have \( 0 \div \left(y - \frac{1}{6} \right) \). Here, \( y - \frac{1}{6} \) is a simple polynomial expression. It's important to understand that simplifying rational expressions usually involves factoring polynomials and reducing them to their simplest form. Though in our exercise the numerator is zero, which simplifies the division instantly, typically you would check both the numerator and the denominator for common factors to simplify.

simplifying expressions

Simplifying expressions in algebra involves reducing them to their most concise form while keeping the equations equivalent. In the given exercise, simplifying \( 0 \div \left(y - \frac{1}{6} \right) \) quickly results in zero using basic division rules. For more complex expressions, you often perform steps like combining like terms, factoring, and cancelling out common factors. For instance, if you had a rational expression like \( \frac{x^2 - 9} {x^2 - 3x} \), you'd factor both the numerator and the denominator and then reduce it by cancelling out the common terms.