Problem 57 - Prealgebra and Introductory Algebra

Question

Is the fifth power of \(-18\) positive or negative?

Step-by-Step Solution

Answer

The fifth power of -18 is negative.

Step 1: Identify the nature of power

The exercise involves raising a number, in this case -18, to the fifth power. Five is an odd number which is important to note when dealing with negative numbers.

Step 2: Apply the principle for odd powers

When a negative number is raised to an odd power, the result remains negative. Thus, (-18)^5 will be negative.

Key Concepts

Odd Powers

When we talk about odd powers, we're specifically referring to raising numbers to powers that are odd numbers, like 1, 3, 5, 7, and so on.
Understanding the behavior of odd powers is crucial when certain numbers, especially negative ones, are involved.
The key characteristic of odd powers is that they maintain the sign of the base number:

• If you raise a negative number to an odd power, the result will be negative.
• If you raise a positive number to an odd power, the result stays positive.

This behavior occurs because multiplying a negative number by itself an odd number of times means the negative sign does not get canceled out. So, remember:

• For negative bases and odd exponents, the outcome will remain negative.

This rule helps us quickly determine the sign of the result without doing the full computation when dealing with powers.

Negative Numbers

Negative numbers are numbers with a value less than zero.
They are represented with a minus sign in front, such as \(-1, -5, -18\), representing values below zero.
When working with negative numbers in mathematical operations such as exponents, the rules of arithmetic become essential.

• Multiplying two negative numbers results in a positive number.
• Multiplying a negative number by a positive number gives a negative product.

These rules are important in understanding how exponents impact negative numbers.
A special rule for exponents is that if a negative number is raised to an odd power, the result will always be negative, as with the example of \((-18)^5\) yielding a negative outcome.
This understanding helps us predict the nature of equations quickly.

Integer Exponents

Exponents, or powers, indicate how many times a number, the base, is multiplied by itself. Integer exponents can be whole numbers that are positive, negative, or zero.
When we deal with integer exponents:

• A positive exponent indicates standard multiplication of the base.
• A zero exponent means the base equals one, provided the base is not zero.
• A negative exponent represents the reciprocal of the number with a positive exponent.

In our exercise with \((-18)^5\), the exponent is 5, a positive integer. This denotes multiplication of the base five times, maintaining the original base's sign since this integer is odd.
Understanding these rules helps with solving various algebraic problems, as they dictate not only the number of multiplications but also the sign and magnitude of the outcome.