Question

Find the value of each expression. \([-5-(-6)]^{2}+(4-2)^{2}\)

Short answer

The value of the expression is 5.

Step-by-step solution

Identify and Simplify Inside the Brackets

First, simplify the expressions inside each set of brackets. For \(-5-(-6)\), this becomes \(-5 + 6 = 1\). For \(4-2\), the solution is \(2\). Now we have the expression \([1]^{2} + [2]^{2}\).

Square the Results

Now, square the results obtained in the previous step. The square of \(1\) is \(1^2 = 1\). The square of \(2\) is \(2^2 = 4\).

Add the Squared Values

Sum the squared values obtained. \(1 + 4 = 5\).

Key concepts

Order of Operations

When working with algebraic expressions, knowing the order of operations is crucial. This set of rules dictates the sequence in which operations must be carried out to accurately solve a mathematical expression. The common acronym to remember this is PEMDAS:

• Parentheses
• Exponents
• Multiplication and Division (from left to right)
• Addition and Subtraction (from left to right)

In the given exercise, we first prioritize operations inside the brackets as per the rule for parentheses. Next, we handle any exponents before moving on to addition or subtraction. This ensures that our calculations are accurate and logically structured. Remember, skipping or misordering any of these steps may lead to incorrect results.

Simplifying Expressions

Simplifying expressions is all about making them as manageable as possible without changing their value. In algebra, this often involves resolving expressions within brackets and combining like terms.

In our original problem, we started by simplifying the calculations inside the brackets. Here we transformed \(-5 - (-6)\) into \(-5 + 6\), which equals 1. Similarly, simplifying \(4 - 2\) results in 2.

By breaking down expressions into simpler parts, we make subsequent calculations easier. This approach significantly reduces the chances of errors and aids in a clearer understanding of the expression as a whole.

Squares and Exponents

Squares and exponents are a fundamental part of algebra that involve multiplying a number by itself. An exponent denotes how many times a number is to be multiplied. When a number has an exponent of 2, it's known as "squaring the number."

In the exercise, once we simplified the expressions in the brackets, we squared the results. The number 1, when squared, remains 1 because \(1^2 = 1\). Squaring the number 2 gives \(2^2 = 4\).

Understanding squares is very helpful, especially in geometry and algebra, as it allows us to calculate area and solve quadratic equations, among other applications. It's important to ensure precise calculation with exponents to avoid growing mistakes in further operations.