Problem 25 - Algebra 1: Concepts and Skills

Question

Evaluate the power. \(1^{8}\)

Step-by-Step Solution

Answer

The result of \(1^{8}\) is 1.

Step 1: Identify the base and the exponent

In the expression \(1^{8}\), the base is 1 and the exponent is 8. You are required to find the product of multiplying 1 by itself 8 times.

Step 2: Apply the rule of exponents

The rule of exponents for any non-zero number \(a\) is: \(a^n = a*a*...*a\) (with \(n\) factors of \(a\)). If \(a\) is 1 then the expression \(1^n\) is always 1 because the product of any number of 1s is always 1.

Step 3: Calculate the power

Substitute the base and the exponent into the rule of exponents. We get, \(1^{8} = 1*1*...*1 = 1\). So, \(1^{8}\) equals 1

Key Concepts

Base and Exponent

When you encounter expressions like 1^{8}, you're looking at a number in a format where it's been raised to a power. The number that is being raised is called the base, and in this example, the base is 1. The power or the exponent is 8, which tells you how many times the base is multiplied by itself. In algebra, understanding the definition and relationship between the base and the exponent is crucial because it sets the stage for more complex operations involving powers.

Here's a quick tip: The base can be any number, positive or negative, while the exponent is typically a positive whole number in basic cases. However, exponents can also be zero, negative, or even fractions, which dictate a different set of rules. The base and exponent together form a power, such as a^n, which plays a pivotal role in algebraic functions and equations.

Rule of Exponents

The 'rule of exponents' refers to the guidelines that govern the operations on powers. One of the fundamental rules is when you're multiplying the base by itself as many times as indicated by the exponent. The expression a^n, as discussed earlier, is essentially the base a multiplied by itself n times.

Among these rules, there's a special case where any non-zero number raised to the power of zero equals 1, written as a^0 = 1. Turning to our exercise, the rule simplifies to 1^n = 1, because multiplying one by itself any number of times will always result in one. This property is not just mathematical trivia; it's a cornerstone concept that can simplify complex algebraic equations involving powers.

Evaluating Powers

Evaluating powers is the process of calculating the result of an exponent expression. It's taking the base and exponent we've discussed and doing the actual math. As you come across different powers in algebra, it’s important to remember that not all evaluations are as straightforward as the example 1^{8}. Here, since we're dealing with the base of 1, any exponent will yield 1 as a result because 1 multiplied by itself any number of times remains 1.

In general, to evaluate powers, one would perform the multiplication as indicated by the exponent the requisite number of times. However, there are shortcuts and patterns you'll begin to notice as you work with exponents more. For instance, any base raised to the power of 2 is called a 'square', and when raised to the power of 3, it's called a 'cube'. Recognizing these patterns helps in quickly evaluating powers without lengthy multiplication.