Problem 1 - Algebra 2 and Trigonometry

Question

What value of \(a\) makes the equation \(6^{a}=1\) true? Justify your answer.

Step-by-Step Solution

Answer

The value of \(a\) that makes the equation true is 0.

Step 1: Understand the Equation

We are given the equation \(6^{a} = 1\). This means that we need to find the value of \(a\) for which raising 6 to the power of \(a\) equals 1.

Step 2: Recall the Rule of Exponents

According to the rule of exponents, any non-zero number raised to the power of 0 is equal to 1. Mathematically, this is expressed as \(b^{0} = 1\) for any \(b eq 0\).

Step 3: Apply the Rule

Using the exponent rule, set the exponent \(a\) equal to 0, because \(6^{0}=1\). Thus, if \(a=0\), then \(6^{a}=6^{0}=1\).

Step 4: Solution Verification

If \(a = 0\), substitute back into the equation to check: \(6^{0} = 1\), which is indeed true. This verifies that our solution is correct.

Key Concepts

Understanding the Rules of Exponents

Exponents play a key role in simplifying mathematical equations and expressions. Understanding the rules of exponents helps in solving exponential equations effectively. One of the fundamental rules is:

• Zero Exponent Rule: Any non-zero number raised to the power of zero results in 1. This can be stated as: if \( b eq 0 \), then \( b^0 = 1 \).

This rule is critical because it allows us to simplify expressions where the base is raised to the zero power. For example, in any case, such as \( 6^0 \) or \( 100^0 \), the result will always be 1 as long as the base isn’t zero.
Understanding these rules is essential when solving equations like \( 6^a = 1 \). Realizing \( a \) must be 0 because \( 6^0 = 1 \) simplifies the problem significantly.

Exploring Exponentiation

Exponentiation is a shorthand for repeated multiplication of a number by itself. Traditionally, it follows the form \( b^n \), where:

• b is the base.
• n is the exponent.

For example, \( 3^4 \) is calculated as \( 3 \times 3 \times 3 \times 3 = 81 \). Every time the exponent multiplies the base by itself as many times as the exponent's value.
When you have an exponent of zero, like in our original exercise, the operation results in 1. So, \( 6^0 \) is not the same as multiplying six by itself zero times, but rather it directly turns into 1 by definition of the zero exponent rule.
Understanding this nuance helps when working with exponentiation, especially in solving different types of exponential equations.

Approaching Equation Solving with Exponents

Solving equations involving exponents requires a good grasp of exponent rules and properties of exponentiation. In the provided problem, \( 6^a = 1 \), we need to determine the value of \( a \) that makes this statement true.

• First, recognize the right side of the equation as a perfect condition for the zero exponent rule.
• Realize that for \( 6^a \) to be equal to 1, \( a \) must be 0 because any number raised to zero equals one.

Setting \( a = 0 \) solves the equation. This step-wise approach ensures the equation balances perfectly. It’s crucial as the verification process, double-checking by plugging \( a = 0 \) back:\[ 6^0 = 1 \]This confirms that the solution is accurate.