Problem 28 - College Algebra with Modeling and Visualization

Question

Evaluate by hand. $$ 5-(-4)^{3}-(4)^{3} $$

Step-by-Step Solution

Answer

The value of the expression is 5.

Step 1: Evaluate the Exponents

First, we need to calculate the exponents \((-4)^{3}\) and \(4^{3}\). Calculate: \((-4)^{3} = (-4) \times (-4) \times (-4) = -64\), Calculate: \(4^{3} = 4 \times 4 \times 4 = 64\).

Step 2: Write the Expression with Calculated Exponents

Substitute the calculated values into the original expression. The expression becomes \(5 - (-64) - 64\).

Step 3: Simplify the Expression

Simplify the expression step-by-step as follows:\(5 - (-64) = 5 + 64 = 69\).Now the expression is \(69 - 64\).

Step 4: Complete the Calculation

Subtract 64 from 69:\(69 - 64 = 5\).

Key Concepts

Exponents

Exponents are a fundamental concept in algebra that indicate how many times a number, known as the base, is multiplied by itself. In simpler terms, if we have the expression \(a^n\), it means \(a\) is multiplied by itself \(n\) times. Understanding and evaluating exponents is crucial for simplifying various mathematical expressions.

Let's take an example from the exercise:

• \((-4)^3\): Here, \(-4\) is the base and \(3\) is the exponent. So you multiply \(-4\) three times: \((-4) \times (-4) \times (-4) = -64\).
• \(4^3\): In this case, \(4\) is the base and \(3\) is the exponent. This results in \(4 \times 4 \times 4 = 64\).

Understanding this process clearly helps when tackling any algebraic problem involving powers. Always remember to deal with exponents before moving on to other operations as per the order of operations rules.

Simplifying Expressions

Simplifying expressions means reducing them to their simplest form without changing their value. This often involves combining like terms and performing operations in a specific order to make expressions easier to understand and solve.

In our example, once the exponents are evaluated, the expression becomes simpler and easier to handle:

• The expression from the exercise becomes \(5 - (-64) - 64\).
• The next step is to simplify by handling the subtraction of a negative number, which turns into addition; thus, \(5 - (-64)\) becomes \(5 + 64 = 69\).
• Now, the expression is further reduced to \(69 - 64\).

By simplifying, the expressions become more straightforward to evaluate. It helps in accurately arriving at the answer while applying rules of arithmetic and algebra systematically.

Arithmetic Operations

Arithmetic operations are basic mathematical calculations which include addition, subtraction, multiplication, and division. Each of these operations follows a specific set of rules and can often be used in combination to solve more complex problems.

In our exercise, we performed a few simple arithmetic operations to resolve the final expression:

• First, dealt with adding: \(5 + 64 = 69\).
• Then, used subtraction: \(69 - 64 = 5\).

These operations are crucial whenever working with algebraic expressions, especially after simplifying complex terms. Understanding the correct sequence of these operations, as guided by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right)), helps you carry out the calculations systematically, leading to accurate results.