Problem 77 - Algebra 2

Question

Simplify each expression. \(\frac{k+3}{5 k \ell} \cdot \frac{10 k \ell}{k+3}\)

Step-by-Step Solution

Answer

The simplified expression is 2.

Step 1: Identify the expressions to multiply

We have two fractions: \( \frac{k+3}{5k\ell} \) and \( \frac{10k\ell}{k+3} \). We need to multiply these fractions together.

Step 2: Multiply the numerators

The numerators of the fractions are \( (k+3) \) and \( 10k\ell \). Multiply these together: \((k+3) \times 10k\ell = 10k\ell(k+3)\).

Step 3: Multiply the denominators

The denominators of the fractions are \( 5k\ell \) and \( (k+3) \). Multiply these: \(5k\ell \times (k+3) = 5k\ell(k+3)\).

Step 4: Simplify the expression

The product of the fractions is \( \frac{10k\ell(k+3)}{5k\ell(k+3)} \). Notice that \((k+3)\) is a common factor in both the numerator and the denominator, so you can cancel it out.

Step 5: Simplify further by canceling terms

After canceling \( (k+3) \), the expression simplifies to \( \frac{10k\ell}{5k\ell} \). Then, \( k\ell \) is also a common factor and can be canceled out.

Step 6: Final simplification

After all cancellations, the expression simplifies to \( \frac{10}{5} \), which simplifies to 2.

Key Concepts

Understanding Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations. In general, they can include:

• Numeric constants, like 5 or 10
• Variables, such as \( k \) or \( \ell \)
• Operations, like addition, subtraction, multiplication, or division

In the exercise, each fraction has an algebraic expression in its numerator and denominator. This means we need to manage these expressions carefully to simplify the fraction correctly.
Understanding how to manipulate and simplify algebraic expressions is essential for solving many algebra problems, including this exercise.

Multiplying Fractions in Algebra

When multiplying fractions, the process is straightforward. Multiply the numerators together to form a new numerator and multiply the denominators together to form a new denominator.
In the given exercise, we have two fractions: \(\frac{k+3}{5k\ell}\) and \(\frac{10k\ell}{k+3}\). To multiply them, we follow these steps:

• Multiply the numerators: \((k+3) \times 10k\ell = 10k\ell(k+3)\)
• Multiply the denominators: \(5k\ell \times (k+3) = 5k\ell(k+3)\)

This process results in a new fraction: \(\frac{10k\ell(k+3)}{5k\ell(k+3)}\).
It is crucial to handle each multiplication carefully to ensure that the expressions are correct before simplifying the fraction.

Finding Common Factors

Common factors in fractions are terms that appear in both the numerator and the denominator and can be "cancelled" to simplify the fraction. Identifying these factors is essential because it greatly simplifies the process of fraction simplification.
In our exercise, after multiplying the fractions, we obtain: \(\frac{10k\ell(k+3)}{5k\ell(k+3)}\).
Here, the term \((k+3)\) is common to both the numerator and the denominator. By cancelling out \((k+3)\), we simplify the fraction. It’s also important to note that sometimes multiple terms can be factors, like \(k\ell\), which can also be cancelled in this case.
This process of finding and cancelling common factors is a critical step in reducing fractions.

Cancelling Terms for Simplification

Cancelling terms is the process of removing common factors from the numerator and the denominator to simplify an algebraic fraction.
In the step-by-step solution, we canceled the term \((k+3)\) because it appears both in the numerator and the denominator. After this cancellation, we are left with \( \frac{10k\ell}{5k\ell} \).
Further simplification involves cancelling \(k\ell\), leaving us with \(\frac{10}{5}\). Simplifying further results in the final answer of 2.
Cancelling terms is a powerful technique in algebra that helps to make complex expressions manageable and reveals the simplest form of a fraction.