Question

What is \(1^7 ?\) A. 7 B. -7 C. 0 D. 1

Short answer

D.1

Step-by-step solution

- Understand the Exponentiation

Exponentiation means raising a number to the power of another number. In this case, we need to raise the number 1 to the power of 7.

- Apply the Exponential Rule

Any number raised to the power of 1 remains that number. For instance, \[ a^1 = \text{a} \].

- Calculate the Exponential

Apply the exponential rule: \[ 1^7 = 1 \]. Since any number to the power of 0 is 1, it simplifies directly.

- Find the Correct Answer

Looking at the choices, the correct answer is D. 1.

Key concepts

Understanding Powers

Exponentiation, often known as raising to a power, is a fundamental mathematical operation. It involves taking a base number and multiplying it by itself a specific number of times. In our example, we need to calculate the power of 1 raised to 7. This can be written mathematically as \(1^7\). Here, 1 is the base, and 7 is the exponent.
Understanding powers is essential for solving various mathematical problems and real-world applications.
Key points to remember:

• The base is the number being multiplied.
• The exponent tells us how many times the base is multiplied by itself.

Exponential Rules

When working with exponents, some rules make calculations easier. One fundamental rule is that any number raised to the power of 1 remains unchanged. In symbolic form, this can be written as \(a^1 = a\), where a is any number. For example:

• \( 5^1 = 5 \)
• \( 10^1 = 10 \)

Another key rule is the power of zero: any number raised to the power of zero is equal to 1, written as \(a^0 = 1\). However, please note this rule isn't applicable in our exercise but is useful for broader understanding.

Applying Mathematical Reasoning

Mathematical reasoning allows us to solve problems logically. For the given exercise, we use the exponential rules to find the correct answer. We start by recognizing that \(1^7\) involves raising 1 to the power of 7. According to the rule \(a^1 = a\), any base number raised to the power of 1 is itself. However, we are dealing with \(1^7\).
Using logical thinking, we realize multiplying 1 by itself multiple times doesn't change its value. Therefore, \(1^7 = 1\). This reasoning also applies to any exponent when the base is 1:

• \(1^3 = 1 \times 1 \times 1 = 1 \)
• \(1^100 = 1 \times 1 \times 1 \ldots = 1 \)

So, in the final step of our solution, we identify the correct answer as D. 1.