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Exponents are a powerful mathematical tool used to represent repeated multiplication. They are written in the form of a base, raised to an exponent, denoted as \( b^n \). The base \( b \) is the number that gets multiplied, and the exponent \( n \) tells us how many times the base is used as a factor. For example, \( 2^3 \) means \( 2 \times 2 \times 2 = 8 \). When solving problems involving exponents, it's crucial to carefully examine the base and exponent, as they can significantly affect the outcome of a calculation.

In many mathematical expressions, especially those with the form \( b^n - 1 \), understanding how exponents work helps us uncover patterns and relationships. For instance, in our specific case, choosing a base of \( b = -1 \) with positive integer exponents allows us to systematically evaluate how changes in \( n \) impact the final expression.

Recognizing whether numbers are odd or even is foundational in checking divisibility. An even number is divisible by 2 and ends in 0, 2, 4, 6, or 8. For instance, 4 is even because \( 4 \div 2 = 2 \). Conversely, an odd number isn't divisible evenly by 2 and ends in 1, 3, 5, 7, or 9, like the number 5.

In our exercise, expressions like \( (-1)^n - 1 \) require examining if the resulting number is even or odd. Inside the expression:

• If \( n \) is even, \( (-1)^n = 1 \), leading to \( 1 - 1 = 0 \), an even number.
• If \( n \) is odd, \( (-1)^n = -1 \), resulting in \( -1 - 1 = -2 \), also even.

Thus, understanding odd and even numbers helps in confirming that the expression is divisible by 2 for all positive integers \( n \).

Algebraic expressions combine numbers, variables, and operation symbols. They offer a way to represent mathematical ideas and relationships succinctly. In our problem, the expression \( b^n - 1 \) is algebraic, blending a base raised to a power with a subtraction operation. Evaluating such expressions requires substituting specific values for any variables involved, like \( b \) and \( n \), and performing the indicated operations.

Choosing the correct base quickly becomes crucial. Our earlier solution demonstrated that using a base of 2 did not yield a divisible result. However, switching to \( b = -1 \) resulted in an expression divisible by 2 for any positive \( n \). In these expressions:

• The variable base \( b \) shows flexibility in expression manipulation.
• Substitute different values effectively to determine divisibility.

Mastering algebraic expressions is invaluable for tackling complex problems by allowing strategic substitutions.