Problem 131 - Elementary and Intermediate Algebra

Question

Translate the set of instructions to an expression and then evaluate it. Subtract the sum of \(-9\) and 8 from the product of the cube of \(-3\) and the opposite of 4

Step-by-Step Solution

Answer

The expression evaluates to 109.

Step 1: Interpret the Phrase 'the sum of -9 and 8'

Find the sum of \(-9\) and \(8\). This simply means adding \(-9\) and \(8\) together. Calculate: \(-9 + 8 = -1\)

Step 2: Interpret 'the cube of -3'

Calculate the cube of \(-3\). The cube means raising \(-3\) to the power of 3.Calculate: \((-3)^3 = (-3) \times (-3) \times (-3) = -27\).

Step 3: Find 'the opposite of 4'

The opposite of \(4\) is \(-4\).

Step 4: Calculate 'the product of the cube of -3 and the opposite of 4'

Multiply the result from Step 2 (cube of \(-3\)) with the result from Step 3 (opposite of \(4\)).Calculate: \(-27 \times (-4) = 108\).

Step 5: Create the Expression

The problem states to subtract the result from Step 1 from the result of Step 4. This can be expressed as:\(108 - (-1)\).

Step 6: Evaluate the Expression

Simplify the expression by solving it.Calculate: \(108 - (-1) = 108 + 1 = 109\).

Key Concepts

Understanding Addition in Algebraic Expressions

Addition is a basic operation in algebra. It is crucial when it comes to combining terms and simplifying expressions. In the example given, you need to find the sum of

• -9 and 8

This means you add the two numbers together. Here are some more details:

• Start with the first number: -9
• Add the second number: 8

When adding these numbers, treat negative numbers as movements to the left on the number line and positive numbers as movements to the right. So, adding -9 and 8 is like starting at -9 and moving right 8 places, ending at -1.
If you encounter more numbers or variables, just continue adding using the same rules.
Addition brings values together and helps form expressions that reflect real-world situations or problems.

Subtraction Simplified

In algebraic expressions, subtraction is how we remove values or difference calculation. The step you are looking at reads:

• Subtract the sum from a product

Here's how to understand this:

• The sum has already been calculated as -1.
• The product of the cube of -3 and the opposite of 4 is calculated as 108.

So, instead of removing -1 from 108, Python teaches us that subtracting a negative is the same as adding it. Therefore, subtracting -1 poses as adding 1. This gives: \[ 108 -(-1) = 108 + 1 \]Thus, the expression ends at 109.
Subtraction can be seen as the opposite movement to addition, moving numbers to the left on the number line or decreasing their value.

Understanding Exponents

Exponents are a powerful tool in algebra, representing repeated multiplication. In the exercise:

• The problem asks for the cube of -3, expressed as \((-3)^3\).

This involves multiplying -3 by itself two more times:

• First, -3 times -3 equals 9
• Second, multiply that result, 9, by -3 again

This gives: \[ (-3) \times (-3) \times (-3) = -27 \]
Exponents allow us to simplify repetitive multiplication into a concise notation and understand growth or diminishment of numbers rapidly.
They are key in various mathematical equations and real-world applications, including scientific calculations.

The Concept of Opposites in Algebra

Opposites in algebra refer to numbers that are of equivalent magnitude but different signs. This means they are equal distances from zero on a number line but in opposite directions. For instance:

• The opposite of 4 is -4
• The opposite of -9 would be 9

Calculating the opposite helps in balancing and simplifying algebraic equations. In problems like this, the use of opposites can change the direction of quantities, turning additions into subtractions and vice versa.
Remember, finding the opposite just changes the sign, helping in tasks like solving for variables or creating more manageable expressions.