Problem 82 - Elementary Algebra

Question

For exercises 15-100, evaluate. $$ -2-5^{2}-(-1) $$

Step-by-Step Solution

Answer

-26

Step 1: Evaluate the exponent

First, solve the exponent in the expression. Here, calculate the value of \(5^2\). \(5^2 = 25\)

Step 2: Simplify the negative sign

Substitute the value obtained from Step 1 back into the expression and simplify the negative signs. The expression becomes: \(-2 - 25 - (-1)\)

Step 3: Remove parentheses

Since the '-' sign before -1 means adding, we can remove the parentheses: \(-2 - 25 + 1\)

Step 4: Perform addition and subtraction

Add and subtract the numbers from left to right. First, solve \(-2 - 25 = -27\), then \(-27 + 1 = -26\)

Key Concepts

Order of Operations

To solve mathematical expressions accurately, it's crucial to follow a specific order of operations. This sequence is often remembered by the acronym PEMDAS: Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). Following PEMDAS ensures that everyone solves the expression in the same way, leading to a consistent and correct answer.

In our original exercise, we followed this order by first evaluating the exponent before addressing other operations. Ignoring the order of operations can lead to incorrect results. Always check which part of the expression needs to be solved first.

Exponents

Exponents are a way to express repeated multiplication of the same number. For example, in the expression \(5^2\), the 2 is an exponent. It tells us to multiply 5 by itself: \(5 \times 5 = 25\).

When dealing with exponents in an expression, evaluate them early as dictated by the PEMDAS rule. In the given problem, the expression \(-2 - 5^{2} - (-1)\) includes \(5^{2}\), which means \(5 \times 5 = 25\). After this simplification, the expression becomes \(-2 - 25 - (-1)\).

Working with exponents first simplifies the expression and makes solving it easier.

Negative Numbers

Negative numbers can be tricky, especially when multiple negative signs are involved. In our given expression, we see several negative signs: \(-2 - 25 - (-1)\). To handle these correctly, remember that subtracting a negative is the same as adding a positive: \(-(-1)\) becomes \(+1\).

So in our example, we rewrite the expression as: \(-2 - 25 + 1\). Simplifying negative numbers properly is crucial in obtaining the right answer. Be sure to carefully manage the signs to avoid errors in your calculations.

Arithmetic Operations

Arithmetic operations are the basic calculations we perform on numbers: addition, subtraction, multiplication, and division. In our example \(-2 - 25 + 1\), we perform both addition and subtraction. According to PEMDAS, these operations are performed from left to right.

First, you solve \(-2 - 25 = -27\), then proceed to adding 1: \(-27 + 1 = -26\). Simplifying step by step in this manner ensures you handle each operation correctly and reach an accurate outcome. Always work through expressions methodically, following the order of operations closely.