Problem 73 - Elementary Algebra

Question

Simplify \(\left|-\left(4^{2}+2^{2}-3^{2}\right)\right|\).

Step-by-Step Solution

Answer

Answer: The simplified expression is 11.

Step 1: Evaluate numbers inside the parentheses

Let's start by evaluating each part inside the parentheses: \(4^2 = 16\) \(2^2 = 4\) \(3^2 = 9\) Now replace these results inside the parentheses: \(-(16 + 4 - 9)\)

Step 2: Simplify inside the parentheses

Now, we will simplify the expression inside the parentheses by doing the addition and subtraction: \(-(16 + 4 - 9) = -(20 - 9) = -(11)\)

Step 3: Apply absolute value

Finally, we will apply the absolute value to the entire expression. Absolute value is the distance of a number from zero. Therefore, the absolute value of a negative number is its positive counterpart: \(|-11| = 11\) The simplified expression is 11.

Key Concepts

Simplification

Simplification in algebra is the process of condensing an expression into its simplest form. It involves combining like terms, performing arithmetic operations, and reducing expressions to make them easier to manage. In the given exercise, we started by evaluating exponents within the parentheses. Evaluating exponents means raising a base number to the power of an exponent, as seen with the terms like \(4^2\) which equals 16.
We then replaced these values in the expression:

• \(-(16 + 4 - 9)\)

The goal of simplification is to resolve everything inside the parentheses first. By adding 16 and 4, we got 20, and then subtracting 9 resulted in 11. We then worked with a negative sign and finally applied the absolute value to get the positive 11. Simplification often makes expressions more manageable, helping in both understanding and further calculations.

Absolute Value

Absolute value is a key concept in algebra representing the distance of a number from zero on a number line. It always results in a non-negative number, as distance cannot be negative. This concept is particularly useful in real-life applications where only magnitude is considered, ignoring the direction.
Consider the expression \(-11\) from our simplified term. Applying the absolute value operation, the negative sign is disregarded, as we're only interested in how far \(-11\) is from 0. Hence, \(|-11| = 11\).
Absolute value simplifies the expression by highlighting its magnitude, which in this problem helped in finally arriving at the simplified outcome of 11. Understanding the absolute value is crucial in solving problems involving real-world quantities such as distance, speed, or modulus of numbers.

Exponents

Exponents in algebra signify repeated multiplication of a base number and are denoted as \(a^n\), where \(a\) is the base and \(n\) is the exponent. They are crucial for expressing large numbers succinctly or simplifying expressions involving repeated multiplication.
In the exercise given, exponents were used to calculate the squares of numbers:

• \(4^2\), which equals 16
• \(2^2\), which equals 4
• \(3^2\), which equals 9

Evaluating these squares was a necessary step before any further simplification could take place. Exponential expressions are foundational in algebra for exploring polynomial operations, growth patterns, and scientific calculations.
Understanding how to interpret and simplify expressions involving exponents is an essential algebra skill. It enables solving problems more efficiently and paves the way for tackling complex algebraic equations and expressions.