Problem 8 - Algebra A Combined Function

Question

Evaluate. $$ 8^{1} $$

Step-by-Step Solution

Answer

The value is 8.

Step 1: Understand the Expression

This expression is an exponentiation problem: \[ 8^1 \] where 8 is the base, and 1 is the exponent.

Step 2: Apply the Power of One Rule

Recall that any number raised to the power of 1 is the number itself. This is known as the power of one rule. Applying this to our expression, we have \[ 8^1 = 8 \].

Key Concepts

Power of One Rule

When dealing with exponents, there are some fundamental rules to remember, and one of these is the **power of one rule**. This rule states that any number raised to the power of one is equal to itself. Simply put, the exponent of 1 means that the base number appears only once in the multiplication. If you see an expression like this:

• \(a^1\)

It evaluates to:

• \(a\)

For example, let's take the number 8: when we raise it to the power of 1, \(8^1 = 8\), because we only have one 8 involved. This rule helps simplify calculations quickly, saving us from unnecessary multiplication.

Base and Exponent

An exponentiation expression is composed of two main components: the **base** and the **exponent**. Understanding these components is crucial to dealing with exponents.- **Base**: This is the number that is being multiplied.- **Exponent**: Indicates how many times to multiply the base by itself.In our original exercise \(8^1\):

• The number 8 is the base.
• The number 1 is the exponent.

Always remember that the base is the number at the bottom and the exponent is the small number written at the top right of the base. Familiarizing yourself with these terms helps simplify complex mathematical expressions and expressions in general mathematics.

Evaluating Exponents

**Evaluating exponents** involves calculating the value of the base raised to the power of the exponent. This concept may seem complex at first, but it's all about repeating multiplication the number of times dictated by the exponent.Here’s a simple way to visualize it:

• When you see \(b^n\), think of multiplying the base \(b\) by itself \(n\) times.
• For instance, for \(8^1\), we multiply 8 by itself just once, resulting in 8.

No actual multiplication is needed with an exponent of 1, as the base stays unchanged. Evaluating exponents like \(8^1\) helps strengthen your understanding of mathematical operations and the role of exponents in maths.