Problem 62 - Algebra 2

Question

State whether each statement is true or false when \(n=1\). Explain. \(2^{n}-1\) is prime.

Step-by-Step Solution

Answer

The statement is false when \(n=1\).

Step 1: Substitute the Value of n

Start by substituting the given value of \(n = 1\) into the expression \(2^n - 1\). This would lead us to \(2^1 - 1\).

Step 2: Simplify the Expression

Now simplify the expression \(2^1 - 1\). Calculate \(2^1\) which equals 2, so \(2^1 - 1 = 2 - 1 = 1\).

Step 3: Determine If Result is Prime

Check if the result from step 2, which is 1, is a prime number. A prime number is defined as a natural number greater than 1 that has no divisors other than 1 and itself. Since 1 is not greater than 1, it cannot be a prime number.

Step 4: Conclude the Truth of the Statement

Since the outcome \(1\) is not a prime number, the statement 'when \(n=1\), \(2^n - 1\) is prime' is false.

Key Concepts

Exponentiation

Exponentiation is a mathematical operation that involves raising a base number to the power of an exponent. The expression \(2^n\) is an example of this concept. Here, the number 2 is the base, and \(n\) is the exponent. Exponentiation is repeated multiplication of the base by itself. For instance, \(2^3\) means multiplying 2 by itself three times, resulting in 8.

In the context of the exercise, we substitute \(n = 1\) into \(2^n\), resulting in \(2^1\). The operation of raising any number to the power of 1 results in the number itself. So, \(2^1 = 2\).

• When \(n = 0\), \(2^n\) would be 1, because any number raised to the power of 0 is 1.
• For negative exponents, such as \(2^{-1}\), we compute the reciprocal: \(1/2\).

Understanding exponentiation is crucial for simplifying expressions in mathematical problems.

Mathematical Expressions

Mathematical expressions combine numbers, variables, and operations to represent a value or relationship. The expression \(2^n - 1\) consists of two primary parts: an exponentiation and a subtraction.

To evaluate this expression when \(n = 1\), we follow these steps:

• First, apply the exponentiation: calculate \(2^1\) which equals 2.
• Next, perform the subtraction: take 2 from the result and subtract 1.

After these calculations, \(2^1 - 1\) simplifies to 1.

Breaking down expressions into smaller parts helps us handle complex problems step-by-step. This structure ensures that you maintain the correct order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction).

Through understanding and practice, mathematical expressions become more intuitive to work with.

Number Theory

Number theory focuses on the properties and relationships of numbers, especially integers. Within it, prime numbers hold significant importance. A prime number is a positive integer greater than 1 with no divisors other than 1 and itself.

In this exercise, we evaluate whether the result of \(2^n - 1\) when \(n=1\) is prime. After the calculation, we found that \(2^1 - 1 = 1\).

It is important to note that 1 is not considered a prime number because it does not meet the basic definition: being greater than 1, and having exactly two distinct positive divisors. To be prime, a number must be indivisible by any other number except for 1 and itself. Examples of prime numbers include 2, 3, 5, and 7.

Understanding these properties helps in various areas of math, including encryption, coding theory, and even puzzles. Number theory enriches our understanding of the foundational elements that constitute the mathematical universe.