Problem 36 - Intermediate Algebra

Question

Simplify each of the numerical expressions. $$ -4(-1)^{2}-3(2)^{3} $$

Step-by-Step Solution

Answer

The simplified expression is -28.

Step 1: Calculate the Power of (-1)^2

First, evaluate the exponentiation \((-1)^2\). Since any number raised to an even power results in a positive number, \((-1)^2 = 1\).

Step 2: Calculate the Power of (2)^3

Next, evaluate \(2^3\). This involves multiplying 2 by itself three times: \(2 \times 2 \times 2 = 8\).

Step 3: Substitute Values into the Expression

Replace the powers in the original expression with the results you calculated:-4 \((-1)^{2}\) - 3 \((2)^{3}\) becomes -4 \(\times 1\) - 3 \(\times 8\).

Step 4: Evaluate the Multiplications

Now, perform the multiplications: -4 \(\times 1 = -4\) and 3 \(\times 8 = 24\).

Step 5: Simplify the Expression

Combine the results of the multiplications by subtracting: -4 - 24 = -28.

Key Concepts

Exponents

Exponents represent the idea of multiplying a number by itself a certain number of times. When you see an expression like

• \((-1)^2\), read it as "negative one squared." This means \(-1\) is multiplied by itself: \((-1) \times (-1)\).
• Since multiplying two negative numbers results in a positive number, \((-1)^2 = 1\).

Similarly, for

• \(2^3\), you're asked to multiply 2 by itself three times: \(2 \times 2 \times 2\). Each successive multiplication helps the number grow rapidly in size.
• Here, \(2^3 = 8\).

Exponents are powerful tools in math, simplifying repeated multiplication into a neat package.

Order of Operations

When simplifying expressions such as

• \(-4(-1)^2 - 3(2)^3\), it's essential to follow the order of operations to get the correct answer.

This order, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right)), dictates the sequence of operations. Given this rule,

• you calculate the powers first, before tackling multiplications and subtractions.

Following PEMDAS ensures that each step is performed in the proper order, preventing mistakes and helping you reach the correct result.

Negative Numbers

Understanding negative numbers is crucial in mathematics, especially when combined with operations like multiplication.

• In the expression \(-4(-1)^2 - 3(2)^3\), the negative numbers influence the outcome of arithmetic operations.
• Multiplying a negative number by a positive number results in a negative product, as seen in \(-4 \times 1 = -4\).
• Similarly, \(-3 \times 8\) gives -24; thus, the subtraction reflects the negative direction in value.

When combining negative numbers with other operations, carefully track whether you're adding or subtracting to maintain accuracy in the final expression.