Problem 57 - Beginning and Intermediate Algebra

Question

Simplify each expression. (Remember the order of operations.) See Examples 4 and 5. $$ |-3|+2^{2}+[-4-(-6)] $$

Step-by-Step Solution

Answer

9

Step 1: Simplify Inside the Absolute Value

Start by simplifying the expression inside the absolute value. The absolute value of \(-3\) is \(|-3| = 3\). This simplifies the expression to: \[3 + 2^2 + [-4 - (-6)]\]

Step 2: Simplify the Exponentiation

Next, evaluate the exponent in the expression. \(2^2\) equals \(4\). Substitute this value in the expression: \[3 + 4 + [-4 - (-6)]\]

Step 3: Simplify Inside the Brackets

Now, simplify inside the brackets. The expression inside the square brackets is \([-4 - (-6)]\). This simplifies to \([-4 + 6]\), which further simplifies to \(2\). Substitute this result into the expression: \[3 + 4 + 2\]

Step 4: Perform Addition

Lastly, perform the addition from left to right. Add \(3\) and \(4\) to get \(7\), and then add \(2\) to get a final result of \(9\). Therefore, the simplified expression is \(9\).

Key Concepts

Absolute Value

Absolute value refers to the distance of a number from zero on the number line, ignoring its sign. This means whether the number is positive or negative, the absolute value is always non-negative.

For instance, if we take \(-3\), its absolute value is \(|-3| = 3\). Similarly, for \(5\), \(|5| = 5\). In the given problem, the expression involves \(-3\) as part of the calculation. We first convert it using the absolute value steps, changing it to \(3\).

This is an essential step before moving on to other operations to ensure calculations incorporate only non-negative numbers.

Exponentiation

Exponentiation is a mathematical operation involving two numbers, the base and the exponent. The exponent tells us how many times to multiply the base by itself.

In our example, we have \(2^2\). The base here is \(2\) and the exponent is also \(2\). Therefore, we multiply \(2 \times 2\), which equals \(4\).

This operation is crucial to simplify before handling other parts of the expression, like brackets or addition, following the correct order of operations rule.

Addition

Addition is one of the most fundamental arithmetic operations. It involves combining numbers to get a total or a sum.

Within this problem, the operation requires us to add numbers in the expression \(3 + 4 + 2\), obtained after simplifying using absolute value and exponentiation.

The process involves several steps:

• First, add \(3\) and \(4\) to make \(7\).
• Next, add \(2\) to the result, \(7\), to get the final sum of \(9\).

Understanding the order of operations (also known as PEMDAS/BODMAS) ensures the addition is performed in the correct sequence after brackets and exponents are dealt with.