Problem 1 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Rewrite the number without using exponents. $$ (-2)^{3} $$

Step-by-Step Solution

Answer

The expression \((-2)^3\) can be rewritten without using exponents as \((-2)\times(-2)\times(-2)\), which equals -8.

Step 1: Understand the exponent

We have the expression \((-2)^3\). The exponent is 3, which means we need to multiply the base, -2, by itself three times.

Step 2: Expand and simplify the expression

Since we have to multiply the base (-2) three times, we can write it as: \[ (-2)\times(-2)\times(-2) \] Now, let's multiply the first two numbers: \[ (+4)\times(-2) \] Now, multiply the result by the last number: \[ -8 \] So, \((-2)^3 = -8\).

Key Concepts

Exponentiation

Exponentiation is a mathematical operation involving two numbers, the base and the exponent. It tells us how many times the base number is multiplied by itself. For example, in the expression \((-2)^3\), \(-2\) is the base and \(3\) is the exponent. The exponent \(3\) signifies that the base, \(-2\), should be multiplied by itself three times.

This operation can be very useful in simplifying large calculations. Instead of writing \((-2)\times(-2)\times(-2)\), we simply write \((-2)^3\). This short notation allows us to understand repetitive multiplication quickly. Remember, the exponent only refers to how many times we multiply the base by itself, not the base it-self becoming larger. It's like a shorthand for repeated multiplication.

Multiplication

Multiplication involves combining numbers in such a way that it is akin to repeated addition. When you multiply numbers, you add a number to itself a certain number of times. In our example, the string \((-2)\times(-2)\times(-2)\) means \((-2)\) is added to itself three times because of multiplication.

Here are some key ideas to remember when multiplying numbers:

• The order of multiplication doesn't matter (commutative property).
• When you multiply two positive numbers or two negative numbers, the result is positive.
• When you multiply a positive number by a negative number, the result is negative.

Multiplying negative numbers involves a special rule: if you have an even number of negative factors in a row, they cancel out to make a positive. If the number is odd, it remains negative. This rule is crucial when simplifying expressions where signs may flip the answer.

Negative Numbers

Negative numbers are numbers that sit below zero on the number line. They represent quantities less than nothing, like a debt or a temperature below freezing. Understanding how to work with negative numbers is essential in many areas of mathematics.

Let's explore some properties of negative numbers:

• When two negative numbers are multiplied, the negatives cancel out, resulting in a positive product. For example, \((-2)\times(-2) = 4\).
• When a negative number is multiplied by a positive number, the product is negative. For instance, \((-2)\times3 = -6\).
• Negative numbers can also appear in exponents, which flips the process depending on whether the exponent is even or odd.

These rules are vital for both simplifying expressions and predicting the behavior of numbers in equations. With practice, you'll gain familiarity and confidence when operating with negative numbers.