Problem 11 - Algebra 1: Concepts and Skills

Question

Write the product in simplest form. $$ \frac{9 x^{2}}{4} \cdot \frac{8}{18 x} $$

Step-by-Step Solution

Answer

The product in simplest forms is \( x \)

Step 1: Multiplication of numerators and denominators

Multiply the numerators together to get the new numerator and do the same with the denominators. This gives: \[ \frac{9 x^{2}*8}{4*18x} = \frac{72 x^{2}}{72x} \]

Step 2: Simplification

Check for factors that are common in both the numerator and the denominator as we can cancel them out. So, divide the numerator and the denominator by 72x which results to: \[ \frac{x}{1} \]

Step 3: Final Answer

After simplifying, the final answer is x, or \[ \frac{x}{1} \] which is equal to x.

Key Concepts

Multiplying Rational Expressions

To multiply rational expressions, you treat them in a similar manner as you would with simple fractions. The process involves multiplying the numerators (the top parts of the fractions) together to get a new numerator, and doing the same with the denominators (the bottom parts).

Consider our example: when we multiply \(\frac{9 x^{2}}{4}\) and \(\frac{8}{18 x}\), we are essentially multiplying the numbers and the variables separately. For the numbers, it's 9 times 8; for the variable part, it's \(x^{2}\) times 1 because the second numerator doesn't have a variable.

The result after this multiplication is \(\frac{72 x^{2}}{72 x}\), which is the new rational expression that we'll need to simplify next. Remember that when you're multiplying rational expressions, you're allowed to cancel out any common factors between numerators and denominators before multiplying to simplify your work, something we call 'cross-cancellation'.

Algebraic Expression Simplification

Simplifying algebraic expressions is a bit like tidying up a room: you're making it cleaner and easier to understand by getting rid of unnecessary clutter. After multiplying rational expressions, you often end up with an expression that can be simplified.

The goal is to divide both the numerator and the denominator by any common factors they share. In the exercise example, both the numerator and denominator had 72x as a common factor.

Common Factors

A factor is a number or expression that divides into another without leaving a remainder. Since 72x is a factor of both the numerator and the denominator, we can simplify the expression \(\frac{72 x^{2}}{72 x}\) by dividing both parts by 72x. This leaves us with \(\frac{x}{1}\), which is just x. Simplifying makes it easier to work with expressions, especially when we apply this to more complex problems.

Common Factors in Algebra

Identifying common factors in algebra is an important skill that can make solving problems much simpler. Common factors are expressions that can be divided out of both the numerator and the denominator of a rational expression.

When looking at our example, we noticed that 72x was a common factor in both the numerator and the denominator. To simplify, we divided the whole expression by this common factor.

Importance of Finding Common Factors

Finding and canceling out common factors reduces expressions to their simplest form. This doesn't just look neater; it also prevents mistakes in subsequent calculations and makes it easier to see relationships and patterns in algebra. Not to mention, simplifying expressions is often required in intermediate steps of solving equations, making this a fundamental skill in algebra.