Question

Simplify the expression. (Review \(8.3 \text { for } 11.7)\) $$\frac{5}{10 x}$$

Short answer

The simplified expression is \( \frac{1}{2x} \)

Step-by-step solution

Identify the Common Factor

In the given expression \(\frac{5}{10x}\), both 5 (numerator) and 10 (denominator) share a common factor of 5.

Simplify the Fraction

To simplify a fraction means to reduce the fraction to its lowest terms by dividing both the numerator and the denominator by their greatest common factor. In this case, dividing 5 by 5 in the numerator gives us 1, and dividing 10 by 5 in the denominator gives us 2. Therefore, the fraction simplifies to \(\frac{1}{2x}\).

Key concepts

Common Factors

Understanding common factors is crucial in simplifying algebraic expressions. A common factor is a number that divides exactly into two or more other numbers. For example, if we look at the numbers 8 and 12, both can be divided by 2 and 4, so these are common factors of 8 and 12.

In the context of algebraic expressions, common factors include numbers, variables, or combinations of both that are present in each term. Recognizing common factors allows us to break down complex expressions into simpler parts. This step makes it easier to manage the algebra involved and pave the way for further simplification processes, such as reducing fractions or identifying the greatest common factor.

Reducing Fractions

The process of reducing fractions is a fundamental skill in algebra. To reduce a fraction, you're looking to find the simplest form where the numerator (the top number) and the denominator (the bottom number) are as small as possible but still maintain the same value of the original fraction.

A reduced fraction is achieved by dividing both the numerator and denominator by their common factors. Importantly, this doesn't change the value of the fraction; it just presents it in a more streamlined form, often making the subsequent calculations clearer and simpler.

Greatest Common Factor

The greatest common factor (GCF), sometimes known as the greatest common divisor, is the largest factor that divides two or more numbers. Determining the GCF is an essential step in many algebraic procedures, including simplifying fractions.

To find the GCF of two numbers, you can list out the factors of each number and choose the largest one they have in common. Alternatively, you can use prime factorization or employ the Euclidean algorithm, which is a more advanced method especially useful for larger numbers. The GCF aids in simplifying expressions to their lowest terms, thus making algebraic operations a lot more straightforward.

Simplifying Fractions

When we talk about simplifying fractions, we refer to the process of making a fraction as simple as possible without changing its value. Simplification typically involves finding the greatest common factor of the numerator and denominator and then dividing both by that number.

The action of simplifying can significantly ease further operations. For instance, when adding, subtracting, or comparing fractions, it's much easier to work with simplified ones. Teaching students how to simplify fractions is critical, as it lays the ground for more advanced aspects of mathematics, including algebra and calculus.