Question

Evaluate the expression. $$\frac{13-4}{18-4^{2}+1}$$

Short answer

The simplified form of the expression is 3.

Step-by-step solution

Simplify the Numerator and the Exponent

Calculate the value of the numerator which is a subtraction operation, and simplify the exponent in the denominator which is raised to the power of 2. Using number operations we find: Numerator: \(13-4 = 9\) Exponent in the denominator: \(4^2 = 16\)

Simplify the Denominator

We replace these calculated values into the equation and complete the subtraction and then addition in the denominator from left to right: Denominator: \(18 -16 + 1 = 3\)

Divide Numerator by Denominator

We have simplified the expression to a simple division, now divide the numerator by the denominator: \(\frac{9}{3} = 3\)

Key concepts

Order of Operations

Understanding the order of operations is crucial when solving mathematical problems, especially when the problem involves multiple steps and operations. This concept is the set of rules that dictates the sequence in which operations should be performed to correctly evaluate a mathematical expression.

When simplifying an expression, the order to follow is often remembered by the acronym PEMDAS: Parentheses, Exponents, Multiplication and Division (from left to right), and then Addition and Subtraction (from left to right). Remember that operations inside parentheses or brackets are always done first, followed by exponents. Only after these steps should you perform multiplication, division, addition, and subtraction in the order they appear from left to right.

In the exercise given, the expression \(\frac{13-4}{18-4^{2}+1}\) requires us to apply the order of operations. The numerator and the calculations within the denominator (which involve an exponent) are simplified first before we proceed to the division, which is the final operation to obtain the answer.

Numerical Expression

A numerical expression is a mathematical statement that can include numbers, operators, and sometimes variables, and can be simplified or evaluated to find its value. Unlike algebraic expressions, numerical expressions do not contain variables and can be computed to a single numerical result.

These expressions can range from being very simple, with just one or two numbers and a single operation, to very complex, with multiple parts that must be calculated step by step respecting the order of operations.

For instance, in our given exercise, \(\frac{13-4}{18-4^{2}+1}\) is a numerical expression that consists of multiple numbers and operators, including subtraction, exponentiation, and division. It's essential to break down the expression into simpler parts, simplifying first what's inside any parentheses and dealing with exponents before proceeding to the other operations.

Exponents

Exponents are a key concept in mathematics, representing how many times a number, known as the base, is multiplied by itself. An exponent is noted by a small number written above and to the right of the base number. The expression \(b^n\) signifies that the base \(b\) is multiplied by itself \(n\) times.

For example, in our problem, the expression \(4^2\) contains an exponent. This exponent tells us to multiply 4 by itself 2 times, resulting in 16. Properly understanding and simplifying expressions with exponents is an essential step before combining or comparing them with other terms in a numerical expression.

When dealing with numerical expressions that include exponents, always calculate these before performing any addition, subtraction, multiplication, or division that doesn't involve terms within parentheses, according to the order of operations.