Problem 58 - Algebra A Combined Function

Question

Simplify each expression. \(|-2|+6^{2}+(-3-8)\)

Step-by-Step Solution

Answer

The simplified expression is 27.

Step 1: Simplify Absolute Value

Begin by simplifying the absolute value expression \(|-2|\). The absolute value of a number is its distance from zero on a number line, regardless of direction. Therefore, \(|-2| = 2\).

Step 2: Calculate the Power Term

Next, compute the power term \(6^2\). Raise 6 to the power of 2, which means multiplying 6 by itself: \(6 \times 6 = 36\).

Step 3: Simplify the Parentheses

Simplify the term inside the parentheses \((-3-8)\). First, subtract 8 from -3, which is \(-3 - 8 = -11\).

Step 4: Combine All Terms

Finally, combine all simplified components of the expression: \(|-2| + 6^2 + (-3-8)\) becomes \(2 + 36 - 11\). Add and subtract these values step-by-step: \(2 + 36 = 38\), and then \(38 - 11 = 27\).

Key Concepts

Absolute Value

The concept of absolute value is fundamental to understanding distances in mathematics. Absolute value represents the distance a number is from zero on the number line, disregarding the direction of that number. This means whether the number is positive or negative, its distance value will always be non-negative.
For example:

• The absolute value of -2 is written as \(|-2|\). Since it is 2 units away from zero, \(|-2| = 2\).
• The absolute value of 5 is \(|5|\), which is 5 units away from zero, hence \(|5| = 5\).

Exponentiation

Exponentiation is a mathematical operation involving two numbers, the base and the exponent. It tells us how many times to multiply the base by itself.
For instance, in the expression \(6^2\), the base is 6 and the exponent is 2. This means multiply 6 by itself: \(6 \times 6 = 36\).
This operation is a key component in various mathematical calculations and simplifies expressions by compactly representing repeated multiplication.

• For example, \(3^3\) means \(3 \times 3 \times 3 = 27\).
• Similarly, \(10^4\) means \(10 \times 10 \times 10 \times 10 = 10000\).

Order of Operations

The order of operations is a rule used to clarify which procedures to perform first in a mathematical expression with more than one operation. It's crucial to follow the correct sequence to arrive at the correct answer.
This sequence can be remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division (left to right), Addition and Subtraction (left to right)).
Applying this to the expression:

• First, we handle the absolute value and exponent: \(|-2|\) and \(6^2\).
• Next, simplify inside the parentheses: calculate \(-3 - 8\).
• Finally, we combine these step-by-step.

Arithmetic Operations

Arithmetic operations are basic mathematical operations including addition, subtraction, multiplication, and division. These are fundamental in simplifying expressions and carrying out calculations.
When dealing with the expression \(|-2| + 6^2 + (-3-8)\), you're primarily focusing on addition and subtraction after the initial simplifications.
Here's how it proceeds:

• Start by adding the result of the absolute value to the power term: \(2 + 36 = 38\).
• Then, subtract the simplified result inside the parentheses: \(38 - 11 = 27\).

By understanding and practicing these operations, you'll develop a solid foundation in handling more complex mathematical problems.