Problem 38 - Algebra for College Students

Question

Simplify each of the numerical expressions. $$5(-1)^{3}-(-3)^{3}$$

Step-by-Step Solution

Answer

The expression simplifies to 22.

Step 1: Evaluate Exponentiation

The expression contains two exponentiation operations: 1. \((-1)^3\): Since -1 multiplied by itself three times is -1 (i.e., -1 × -1 × -1 = -1).2. \((-3)^3\): Since -3 multiplied by itself three times is -27 (i.e., -3 × -3 × -3 = -27).

Step 2: Substitute Powers Back into Expression

Substitute the values obtained for the powers back into the original expression. The expression \(5(-1)^{3}-(-3)^{3}\) transforms to:\(5(-1) - (-27)\).

Step 3: Simplify Multiplication

Evaluate the multiplication in the expression: \(5(-1) = -5\).Thus, the expression now becomes:\(-5 - (-27)\).

Step 4: Resolve Double Negative

A negative sign before a positive number indicates subtraction, while two negative signs cancel each other out, making a positive. Therefore: \(-5 - (-27)\) becomes \(-5 + 27\).

Step 5: Perform Final Addition

Add the numbers from the expression:\(-5 + 27 = 22\).Thus, the final simplified expression is 22.

Key Concepts

Exponentiation

Exponentiation is a mathematical operation involving two numbers, the base and the exponent. The exponent indicates how many times the base is multiplied by itself.
For example, in the expression \((-1)^3\), the base is -1, and the exponent is 3.

• This means you multiply -1 by itself a total of three times: \(-1 \times -1 \times -1 = -1\).
• Similarly, for \((-3)^3\), you multiply -3 by itself three times: \(-3 \times -3 \times -3 = -27\).

When dealing with negative bases raised to an odd exponent, the result is negative. Negative numbers multiplied an odd number of times remain negative, as illustrated above. For even exponents, the result would be positive, since negative numbers multiplied an even number of times cancel the negative signs.

Negative Numbers

Understanding negative numbers is crucial in algebra, as they behave differently in various operations. A negative number is simply a number with a minus sign in front of it, indicating it is less than zero. When dealing with expressions involving negative numbers, remember the following:

• The negative of a negative is a positive. For example, in the expression \(-(-27)\), the double negative turns into a positive, resulting in \(+27\).
• When multiplying or dividing two negative numbers, the result is positive. Conversely, multiplying or dividing a positive number with a negative results in a negative.

These principles help simplify expressions correctly, avoiding common pitfalls when combining operations like subtraction and multiplication.

Arithmetic Operations

Arithmetic operations include addition, subtraction, multiplication, and division, and they form the foundation of algebra simplification. Performing these operations requires understanding the rules of precedence and operations involving negative numbers.
In this exercise:

• First, we performed exponentiation, then followed by multiplication: 5 multiplied by -1 equals -5.
• Next, came the subtraction of two expressions: \(-5 - (-27)\).
• We addressed the negative sign leading into the subtraction, transforming it into addition to simplify the expression.
• The final addition \(-5 + 27 = 22\) delivers the simplified result of the expression.

This systematic handling of arithmetic operations ensures that expressions are simplified correctly and logically, leading to accurate results.