Problem 12 - Algebra 1

Question

Find the product. $$-(-1)^{5}$$

Step-by-Step Solution

Answer

The product is 1.

Step 1: Exponentiation

Perform the exponentiation operation inside the parentheses. Any number to the odd power maintains it sign. Hence, \( (-1)^{5} = -1 \)

Step 2: Negation operation

The negation of a number changes the sign of the number. So, the negative sign in front of the parentheses changes the sign of the result we obtained in the previous step. In other words, we take the negation of -1, which becomes positive: \(-(-1) = 1 \).

Key Concepts

Exponentiation

Exponentiation is a fundamental mathematical operation involving two numbers, the base, and the exponent. It is written as \( a^n \), where \( a \) is the base and \( n \) is the exponent. This expression represents the number \( a \) multiplied by itself \( n \) times. For example, \( 2^3 = 2 \times 2 \times 2 = 8 \).
In the given exercise, the base is -1 and the exponent is 5, an odd number.

• When the base is negative and the exponent is an odd number, the resulting product keeps the negative sign. This is because multiplying an odd number of negative values results in a negative product. Thus, \( (-1)^5 = -1 \).
• Remember that if the exponent were an even number, the product would be positive due to the pairing of negative signs, resulting in a positive product.

Understanding the behavior of exponents with negative bases is key to mastering exponentiation.

Negation

Negation in mathematics refers to changing the sign of a number. If the number is positive, its negation is negative, and vice versa.
When using the '-' sign in front of a number or an expression, it indicates negation, which essentially involves multiplying by -1.

• In our exercise, after solving \((-1)^5 = -1\), we are left with the expression \(-(-1)\).
• Applying the negation operation to -1 changes its sign to positive 1. So, \(-(-1) = 1\), as negating a negative effectively "undoes" the negative, turning it into positive.

This operation is vital for solving problems involving integers, especially when dealing with expressions with multiple layers of negative signs.

Odd Exponents

Odd exponents are exponents that are odd integers such as 1, 3, 5, 7, and so on. They fundamentally affect the sign of a product when dealing with negative bases.
An odd exponent on a negative base value retains the negative sign of the base.

• Consider a negative base like -2 raised to an odd exponent, like 3: \((-2)^3 = -2 \times -2 \times -2 = -8\).
• A key insight is that each pair of negative numbers results in a positive product, but since the exponent is odd, there is one unpaired negative, ensuring the final result is negative.

This is a crucial pattern to keep in mind when performing calculations with integer operations, as it governs the final sign of the expression's result.