Problem 23 - Elementary and Intermediate Algebra

Question

Simplify. $$ (-2)^{5} $$

Step-by-Step Solution

Answer

-32

Step 1 - Understand the Exponentiation

Exponentiation means multiplying a number by itself a certain number of times. Here, \((-2)^{5}\) means we multiply -2 by itself 5 times.

Step 2 - Multiply Step by Step

Start by multiplying in pairs: \(-2 \times -2 = 4\). Now multiply 4 by -2 to get \(-8\). Continue by multiplying -8 by -2, resulting in \(( -8 \times -2 = 16)\). Finally, multiply 16 by -2 to get \(( 16 \times -2 = -32)\).

Step 3 - Combine Results

Following the steps one by one, you get \(-2 \times -2 = 4\), then \((4 \times -2 = -8)\), then \(( -8 \times -2 = 16)\), and finally \(( 16 \times -2 = -32)\) which simplifies to -32.

Key Concepts

Multiplication

Multiplication is one of the basic arithmetic operations where you combine equal groups. If you have 3 groups of 4 apples, you multiply 3 by 4 to get 12 apples in total. In the context of our problem, exponentiation is repetitive multiplication. When we see an expression like \((-2)^5\), it means we are multiplying -2 by itself 5 times. Each time you multiply, it’s called an iteration.

Here’s a recap of the process:

• First pair multiplication: \(-2 \times -2 = 4\)
• Second pair multiplication: \(4 \times -2 = -8\)
• Third pair multiplication: \(-8 \times -2 = 16\)
• Last multiplication: \(16 \times -2 = -32\)

As you continue, remember that each step in our iteration either maintains or changes the result's sign.

Negative Numbers

Negative numbers are numbers less than zero, usually denoted by a minus sign ( - ). They have some special rules when it comes to operations like multiplication. In multiplication:

• A negative number multiplied by a positive number gives a negative result. For example, \(-2 \times 3 = -6\)
• A negative number multiplied by another negative number gives a positive result. For example, \(-2 \times -3 = 6\)

Understanding these rules helps us correctly handle situations where we have multiple negative numbers involved, such as with our exponentiation problem. This is why \(-2 \times -2 = 4\) is positive, and why continuing to multiply by -2 alternates the sign of the result.

Powers of Integers

Powers of integers involve raising a base number to an exponent. The exponent tells you how many times to use the base number in a multiplication. In our problem \((-2)^5\), -2 is the base, and 5 is the exponent. When you raise a negative integer to an odd power, the result is negative. When you raise a negative integer to an even power, the result is positive.

This is because:

• Odd powers multiply the base an odd number of times, resulting in a negative sign. \(-(2^5)\) = -32
• Even powers multiply the base an even number of times, cancelling out the negative signs, resulting in a positive sign. \(-(2^4)\) = 16

The rules are consistent and can help us quickly understand the sign of our answer before completing the calculations.