Problem 123 - Intermediate Algebra

Question

Evaluate each expression. $$-3|4-8|+(4+2 \cdot 3)^{3}$$

Step-by-Step Solution

Answer

The value of the expression is 988.

Step 1: Evaluate the Absolute Value

First, evaluate the expression inside the absolute value: \( |4 - 8| \). Calculate \( 4 - 8 \) which gives \( -4 \). The absolute value of \(-4\) is \(4\).

Step 2: Multiply with the Coefficient

Next, multiply the absolute value result by \(-3\): \(-3 \times 4 = -12\).

Step 3: Solve the Multiplication Inside Parentheses

Now look at the expression \(2 \cdot 3\) inside the parentheses: \(2 \times 3 = 6\).

Step 4: Add Inside Parentheses

Add \(4\) and the result from step 3: \(4 + 6 = 10\).

Step 5: Evaluate the Exponentiation

Raise \(10\) to the power of \(3\): \(10^{3} = 1000\).

Step 6: Combine Results

Finally, add the results from Step 2 and Step 5: \(-12 + 1000 = 988\).

Key Concepts

Absolute Value

Absolute value is a concept used in mathematics to denote the distance of a number from zero on the number line. This means, it is simply the non-negative value of a number. For any real number, the absolute value is represented as \(|x|\), where \(x\) is any number.

The absolute value of \(-4\), for example, is \(4\) because the distance from \(-4\) to zero is 4 units. Simply put,

• The absolute value of a positive number is the number itself.
• The absolute value of zero is zero.
• The absolute value of a negative number is its positive equivalent.

Absolute value plays a crucial role in various mathematical expressions, including algebraic expressions like the one in our example. Taking the absolute value is often an important first step in simplifying or evaluating these equations.

Exponentiation

Exponentiation is a shorthand way to express repeated multiplication of the same number by itself. In the expression \( a^n \), \(a\) is the base and \(n\) is the exponent. The exponent tells you how many times the base is multiplied by itself.

For instance, in \(10^3\), the base \(10\) is multiplied by itself three times:

• \(10 \times 10 = 100\)
• Then \(100 \times 10 = 1000\)

Exponentiation is a key part of mathematical problems because it is used to express very large or very small numbers efficiently. In our original exercise, once the parentheses were dealt with, exponentiation was necessary to reach the solution.

Order of Operations

The order of operations is a set of rules that dictates the sequence in which mathematical operations should be performed. Often remembered by the acronym PEMDAS:

• **P**arentheses – First, calculate any expressions inside parentheses.
• **E**xponents – Next, evaluate any exponents.
• **MD** – Multiply or Divide, from left to right.
• **AS** – Add or Subtract, from left to right.

Following this order ensures that everyone evaluates expressions in a consistent manner, avoiding disagreement about the final result. Ignoring the order can lead to errors, especially in complex expressions. In our example, following PEMDAS allowed us correctly to resolve the inner values before advancing to exponentiation, multiplication, and addition.

Integer Operations

Working with integers involves performing addition, subtraction, multiplication, and division on whole numbers, which include positive numbers, negative numbers, and zero. It's essential to understand how these operations affect integers:

• Addition – Adding two positive numbers results in a positive sum, while adding two negative numbers results in a negative sum.
• Subtracting – Subtracting a negative is equivalent to adding its positive counterpart.
• Multiplying – Multiplying two integers with the same sign results in a positive product, while different signs result in a negative product.
• Division – Similar to multiplication, division with the same sign results in a positive quotient, and different signs produce a negative quotient.

Understanding integer operations is crucial in algebraic expressions, as in this exercise. By carefully applying these rules, you can accurately combine results from earlier steps, like multiplying and adding negative and positive numbers to reach the final solution.