Question

EXPRESSIONS WITH FRACTION BARS Evaluate the expression. $$ \frac{75-5^{2}}{11+(3 \cdot 4)} $$

Short answer

\(\frac{50}{23}\)

Step-by-step solution

Simplify the Numerator

In the numerator we have the expression 75 - 5^{2}. We start by calculating the exponent: \(5^2 = 25\). Then, subtract this from 75: \(75 - 25 = 50\). So, the numerator simplifies to 50.

Simplify the Denominator

In the denominator we have the expression 11 + (3 \cdot 4). Start by performing the multiplication: \(3 \cdot 4 = 12\). Then, add this to 11: \(11 + 12 = 23\). So, the denominator simplifies to 23.

Perform the Division

Now that the numerator and denominator have been simplified, perform the division: \(\frac{50}{23}\). This fraction cannot be simplified further, so leave it as it is.

Key concepts

Simplifying Numerators and Denominators

When solving expressions that include fractions, understanding how to simplify the numerator and the denominator is paramount. Simplification involves performing all possible arithmetic operations, such as addition, subtraction, multiplication, and division, as well as applying exponents, to reduce the expression to its simplest form.

To do this properly, you must remember to handle any terms with exponents first before dealing with multiplication, division, addition, or subtraction. For example, if you encounter a term like \(5^2\), it's crucial to calculate the exponent before subtracting it from another number. In the case of our exercise, calculating \(5^2 = 25\) and then subtracting it from 75 results in the simplified numerator of 50. The same logic applies to the denominator, where terms inside brackets should be evaluated prior to operations outside them, leading to \(11 + (3 \cdot 4) = 23\) in our problem. Simplifying both parts of a fraction before attempting to divide them is an effective method to ensure the final answer is in its most reduced form.

Order of Operations

In mathematics, the order in which operations should be performed is governed by a hierarchy known as PEMDAS or BODMAS, depending on the region. This stands for Parentheses (Brackets), Exponents (Orders), Multiplication and Division (from left to right), and Addition and Subtraction (from left to right).

This order of operations is crucial when evaluating algebraic expressions. Neglecting this sequence can lead to incorrect results. When looking at our exercise example, \(\frac{75-5^{2}}{11+(3 \cdot 4)}\), we apply the order of operations by handling the exponent first (\(5^2\)), then multiplication inside the brackets (\(3 \cdot 4\)), and finally performing addition and subtraction at the end. Remember, multiplication and division are on the same level, so if they appear together, just work from left to right, as should be done with addition and subtraction.

Exponents in Algebra

In algebra, exponents are used frequently to express numbers or variables raised to a certain power. The exponent denotes how many times a number called the base is multiplied by itself. For instance, \(5^2\) means \(5 \cdot 5\) which equals 25. The proper handling of exponents is essential when simplifying algebraic expressions.

It's important to remember that when dealing with exponents in algebra, not only do numerical bases come into play, but variables can also be involved. When calculating with exponents, always perform these operations before other arithmetic processes, as per the order of operations. This ensures that the structure of the algebraic expression remains intact throughout the simplification process. For example, in our exercise, the term \(5^2\) must be dealt with before we subtract it from 75 to reach the simplified form of the numerator.