Problem 12 - Algebra 1

Question

Simplify the expression if possible. $$\frac{14 x^{2}}{50 x^{4}}$$

Step-by-Step Solution

Answer

So, the simplified form of the given expression \(\frac{14x^{2}}{50x^{4}}\) is \(\frac{7}{25}x^{-2}\)

Step 1: Break Down the Expression

First, break down the given expression \(\frac{14 x^{2}}{50 x^{4}}\) into two fractions, one for the constants and one for the variables. So it becomes \(\frac{14}{50} \times \frac{x^{2}}{x^{4}}\)

Step 2: Simplify the Constants

The numerical constants 14 and 50 have a common factor of 2, so you can simplify the fraction to \(\frac{7}{25}\). So, now we have \(\frac{7}{25} \times \frac{x^{2}}{x^{4}}\)

Step 3: Simplify the Variables

Next, simplify the variables using the law of exponents which states that for any real number 'a' and any integers m and n, \(a^{m}\div a^{n} = a^{m-n}\). Therefore \(\frac{x^{2}}{x^{4}} = x^{2-4} = x^{-2}\)

Step 4: Final Simplification

Combine the constants and variables again for the final simplified form. So, the simplified version of the given expression is \(\frac{7}{25} \times x^{-2}\)

Key Concepts

Laws of Exponents

The laws of exponents, also known as the rules of exponents, are essential tools in simplifying expressions involving powers. Understanding these rules helps you manage and simplify terms with exponents easily.

One crucial rule is the quotient of powers. This rule states that when you divide two exponents with the same base, you subtract the exponents. In mathematical form, for any nonzero number 'a' and integers m and n, this is expressed as \(a^{m} / a^{n} = a^{m-n}\).

Here's how the quotient of powers rule helps us in simplifying:\

• In the expression \(\frac{x^{2}}{x^{4}}\), both terms have the same base 'x'.
• According to the exponent rule, subtract the exponents: \(2 - 4 = -2\).
• Therefore, the expression simplifies to \(x^{-2}\).

Understanding such exponent rules is crucial for working with algebraic expressions effectively, especially when you're simplifying fractions or handling equations with exponents.

Fraction Simplification

Simplifying fractions involves dividing both the numerator and the denominator by their greatest common factor (GCF). This process reduces the fraction to its simplest form.

In the context of the given expression \(\frac{14}{50}\), let's simplify this using fraction simplification:

• Identify the GCF of 14 and 50, which is 2.
• Divide the numerator and the denominator by their GCF: \(\frac{14}{2} = 7\) and \(\frac{50}{2} = 25\).
• This gives us the simplified fraction \(\frac{7}{25}\).

With this simplified fraction, we can say that reducing fractions to their simplest form ensures that the expression is as compact and easy to understand as possible. This technique is valuable not only in algebraic expressions but in performing operations across mathematics.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations such as addition, subtraction, multiplication, and division. They form the backbone of algebra, enabling us to model and solve problems.

When dealing with algebraic expressions like \(\frac{14 x^{2}}{50 x^{4}}\), one important step is to segregate constants from variables before simplifying, as shown in the initial solution steps.

• Split \(\frac{14 x^{2}}{50 x^{4}}\) into two parts: \(\frac{14}{50}\) and \(\frac{x^{2}}{x^{4}}\).
• Simplify each part separately using appropriate mathematical rules.
• Combine the simplified forms to reach the final expression.

This method of breaking down expressions into simpler parts makes the process more manageable and helps ensure accuracy. Understanding and working proficiently with algebraic expressions is crucial for advancing in algebra and other mathematical disciplines.