Problem 71 - Algebra A Combined Function

Question

Evaluate each expression. $$ 1^{8} $$

Step-by-Step Solution

Answer

The expression evaluates to 1.

Step 1: Understanding Exponentiation

The expression given is \(1^8\), which is an example of exponentiation. In exponentiation, the base number (1 in this case) is multiplied by itself a number of times equal to the exponent (8 in this case).

Step 2: Recognizing the Properties of One

A fundamental property of the number one is that any power of one is always one. This is because multiplying one by itself any number of times still results in one.

Step 3: Applying the Property

Using the property that \(1^n = 1\) for any integer \(n\), the expression \(1^8\) evaluates to 1.

Key Concepts

Powers of One

When we talk about the powers of one, we are focusing on a unique characteristic of the number one. Regardless of how many times one is multiplied by itself, the result is always one. This is due to the multiplicative identity property.

If you take 1 and multiply it by 1 once, twice, or even eight times, the outcome remains unchanged. This means:

• \(1^1 = 1\)
• \(1^2 = 1\times 1 = 1\)
• \(1^3 = 1\times 1 \times 1 = 1\)
• Continue this pattern, and you'll always end with 1.

Therefore, the expression \(1^8\) simply equals 1.

Understanding this concept is crucial because it simplifies calculations and highlights the stable nature of the number one when used in exponentiation.

Base and Exponent

In the world of mathematics, the terms base and exponent are foundational when dealing with exponents. The base is the number that gets multiplied, while the exponent tells us how many times to multiply the base.

In our expression \(1^8\), 1 is the base and 8 is the exponent.

• Base: It's the number you will repeatedly multiply. Here, the base is 1.
• Exponent: This indicates the number of times the base is multiplied by itself. In this case, the exponent is 8.

So, when you see \(1^8\), envision multiplying 1 by itself until you've done so eight times. Recognizing these roles helps decode many exponent problems.

Properties of Exponents

Exponents come with several fascinating properties that make mathematical calculations both simpler and more powerful. These properties guide how we manipulate and understand numbers in exponential form.

Key properties to remember include:

• Multiplying Powers with the Same Base: When you multiply exponents with the same base, add the exponents. For example, \(a^m \times a^n = a^{m+n}\).
• Power of a Power: When you raise a power to another power, multiply the exponents. For example, \((a^m)^n = a^{m \times n}\).
• Zero Exponent Rule: Any nonzero base raised to the power of zero equals one: \(a^0 = 1\). This pigeonholes into the identity and simplification of expressions.
• One as a Base: As covered, any power of one will always be one: \(1^n = 1\).

Each property streamlines complex operations, enabling more efficient problem-solving in both everyday math and advanced computations.