Problem 57 - Algebra A Combined Function

Question

Simplify each expression. \(|-3|+2^{2}+[-4-(-6)]\)

Step-by-Step Solution

Answer

The simplified expression is 9.

Step 1: Simplify the absolute value

The expression contains the absolute value \(|-3|\). Remember, the absolute value of a number is its distance from zero on the number line, which is always positive. So, \(|-3| = 3\).

Step 2: Calculate the exponent

The next part of the expression is calculating the exponent \(2^2\). Remember, \(2^2\) means 2 multiplied by itself, which is \(2 \times 2 = 4\).

Step 3: Simplify the expression inside square brackets

The last part is simplifying \([-4 - (-6)]\). First, deal with the brackets: Subtracting a negative is the same as adding, so \(-(-6)\) becomes \(+6\). Therefore, \([-4 + 6]\) simplifies to \(2\).

Step 4: Add the results together

Now, combine all the simplified parts: \(3 + 4 + 2\). First add \(3 + 4 = 7\), then add \(7 + 2 = 9\). So, the simplified expression is \(9\).

Key Concepts

Absolute Value

Understanding the concept of absolute value is essential to grasping real number operations. Absolute value refers to the distance of a number from zero on the number line, and it is always a non-negative number.

• The absolute value of a positive number is the number itself.
• The absolute value of a negative number is its positive counterpart. For example, \(|-3|\) is 3.
• The absolute value of zero is zero, since it is already at the point of origin on the number line.

Absolute value is denoted by vertical bars around the number, like so: \( |x| \). In our example, this means assessing \( |-3| \) and converting it into \( 3 \). Understanding absolute values is fundamental in simplifying expressions involving mixed numbers.

Exponents

Exponents are a way to express repeated multiplication of the same number and are a core concept in algebra. An exponent tells you how many times to multiply the base number by itself.

• The expression \( 2^2 \) means multiplying the base 2 by itself once, which results in \( 2 \times 2 = 4 \).
• Generally, \( a^n \) means multiplying \( a \) by itself \( n \) times.
• Exponents can be used with positive, negative, and even fractional numbers.

In the context of simplifying expressions, exponents simplify to a single value, which then gets combined with other parts of the expression. In our example, solving \( 2^2 \) gives us a straightforward result that feeds into the overall simplification process.

Arithmetic Operations

Arithmetic operations involve basic calculations like addition, subtraction, multiplication, and division. In simplifying algebraic expressions, you often need to perform a sequence of these operations.
For example, when you have to simplify an expression such as \([-4 - (-6)]\), we apply subtraction and addition:

• The expression within the brackets, \[-(-6)\], indicates subtracting a negative, which essentially turns into addition, \( +6 \).
• Thus, \([-4 + 6]\) simplifies to 2.

Operations often require careful application of the order of operations, ensuring calculations proceed accurately to yield correct results. In the exercise, after simplifying individual parts, results are aggregated, leading to the final simplified expression.

Algebraic Simplification

Algebraic simplification involves reducing expressions to their simplest form by resolving elements one by one. This can include applying absolute values, calculating exponents, and performing arithmetic operations.
The goal is to break down an expression into basic components that can be easily managed:

• Identify and compute absolute values and exponents individually.
• Simplify expressions within brackets or parentheses to one number.
• Combine all parts by following the order of arithmetic operations.

For the given example, the approach involves tackling each part step-by-step: first resolving the absolute value, then the exponent, followed by the operation within brackets, and finally combining these results. This strategic approach ensures the expression is brought to its simplest form, achieving clarity in result.