Question

Find \(3 \times \frac{x}{11(x+y)}\).

Short answer

Answer: The simplified expression is \(\frac{3x}{11(x+y)}\).

Step-by-step solution

Write down the exercise

We start by writing down the expression: \(3\times \frac{x}{11(x+y)}\).

Multiply 3 by the fraction

Now we multiply 3 by the fraction: \(3 \times \frac{x}{11(x+y)} = \frac{3x}{11(x+y)}\).

Check if any further simplification is needed

In this case, there are no common factors between the numerator (3x) and the denominator (11(x+y)), so the final expression is: \(\frac{3x}{11(x+y)}\).

Key concepts

Algebraic Expressions

In mathematics, an algebraic expression is a combination of numbers, variables, and arithmetic operations like addition, subtraction, multiplication, and division. In the exercise Find \(3 \times \frac{x}{11(x+y)}\), we are dealing with an algebraic expression that involves fraction multiplication.

An important skill when working with algebraic expressions is identifying how to combine like terms and how to factor expressions. When you factor, you're looking for expressions that can be rewritten as products of simpler expressions. Although in our given exercise no factoring was necessary, it's crucial to always consider this step as it can greatly simplify the expression.

Fraction Multiplication

When multiplying fractions, you simply multiply the numerators (the top numbers) together and the denominators (the bottom numbers) together. If you have whole numbers and fractions, like in our exercise (\(3 \times \frac{x}{11(x+y)}\)), you treat the whole number as a fraction with the number itself as the numerator and 1 as the denominator. Thus, multiplying 3 and \(\frac{x}{11(x+y)}\) gives us \(\frac{3}{1} \times \frac{x}{11(x+y)} = \frac{3x}{11(x+y)}\).

It's always important to check if you can simplify the fraction after the multiplication. This involves looking for common factors in the numerator and denominator that can be divided out, effectively 'reducing' the fraction to its simplest form.

Simplification of Fractions

The process of simplification of fractions involves reducing fractions to their simplest form where the numerator and the denominator are as small as possible and have no common factors other than 1. To simplify a fraction, you divide both the numerator and the denominator by their greatest common factor (GCF).

In our exercise (\(\frac{3x}{11(x+y)}\)), we checked for the GCF between the numerator and the denominator. Since 3x and 11(x+y) do not share any common factors, the fraction is already in its simplest form. When simplifying algebraic fractions, always be on the lookout for common variables or factors that can be canceled out to achieve the most reduced form of the expression.