Problem 35 - Elementary Algebra

Question

\((-2)^{3}\)

Step-by-Step Solution

Answer

-8

Step 1: Understand the Problem

The given problem is to evaluate the expression \( (-2)^3 \). This means we need to find the result of raising \( -2 \) to the power of 3.

Step 2: Break Down the Exponentiation

Exponentiation means multiplying the base number by itself as many times as the exponent indicates. For \( (-2)^3 \), we need to multiply \( -2 \) by itself 3 times.

Step 3: Perform the Multiplication

Now, multiply \( -2 \) by itself 3 times: \( (-2) \times (-2) \times (-2) \).

Step 4: Calculate the Result

First, calculate \( (-2) \times (-2) \). This results in \( 4 \). Then, multiply the result by \( -2 \): \( 4 \times (-2) = -8 \).

Step 5: Write the Final Answer

The value of \( (-2)^3 \) is \( -8 \).

Key Concepts

negative numbers

Negative numbers are numbers less than zero. They are represented with a minus sign in front of them. For example, -1, -2, and -3 are negative numbers. In arithmetic and algebra, they follow specific rules:

• Adding a negative number is the same as subtracting its positive counterpart. For example, 5 + (-3) is the same as 5 - 3.
• Subtracting a negative number is the same as adding the positive counterpart. For example, 5 - (-3) is the same as 5 + 3.
• When multiplying two negative numbers, the result is positive. For example, (-2) * (-3) = 6.
• When multiplying a positive number and a negative number, the result is negative. For example, 2 * (-3) = -6.

Understanding these rules is important for solving problems involving negative numbers and operations like exponentiation.

powers

Powers, also known as exponents, are a shorthand way to express repeated multiplication of the same number. The notation for powers is given by the form \(a^b\), where \(a\) is the base, and \(b\) is the exponent. This means \(a\) is multiplied by itself \(b\) times.

• For example, \(2^3\) means \(2 \times 2 \times 2\), which equals 8.
• Exponents can also be negative or zero. For instance, \(2^{-3}\) means \(\frac{1}{2^3}\), which equals \(\frac{1}{8}\).
• Any non-zero number raised to the power of zero is 1, i.e., \(a^0 = 1\).

In the context of our problem, \((-2)^3\), we multiply -2 by itself three times.

multiplication

Multiplication is one of the basic arithmetic operations where two numbers are combined to get their product. There are some rules to keep in mind, especially when dealing with negative numbers:

• Multiplying two positive numbers results in a positive product. For example, 2 * 3 = 6.
• Multiplying two negative numbers results in a positive product, because the negatives cancel out. For example, (-2) * (-3) = 6.
• Multiplying a positive number by a negative number results in a negative product. For example, 2 * (-3) = -6.

When we multiply \((-2) \times (-2) \times (-2)\), we first multiply the two negative numbers to get a positive product: \((-2) \times (-2) = 4\). Then, we multiply this result by -2 to get the final product: \(4 \times (-2) = -8\). This follows the rule that a positive number multiplied by a negative number yields a negative result.