Problem 45 - Beginning and Intermediate Algebra

Question

Evaluate. $$ -1^{5} $$

Step-by-Step Solution

Answer

The value of \(-1^5\) is \(-1\).

Step 1: Interpret the Expression

The expression is \(-1^5\). According to order of operations, evaluate the exponent first, and then apply any multiplication or negative signs.

Step 2: Evaluate the Exponentiation First

Evaluate the expression \(1^5\). Any number raised to the 0th power is 1. So, this step simplifies to: \(1^5 = 1\).

Step 3: Apply the Negative Sign

After evaluating the exponentiation, apply the negative sign before the result. So the expression becomes \(-1^5 = -(1^5) = -1\).

Key Concepts

Exponentiation

Exponentiation is a fundamental math concept that involves raising a number, also known as the base, to the power of an exponent. In the expression \[-1^5\], the base is 1 and the exponent is 5. This operation is read as "1 raised to the power of 5."

Here’s how to evaluate it step-by-step:

• The exponent indicates how many times the base is multiplied by itself. So, in this example:\[1^5 = 1 \times 1 \times 1 \times 1 \times 1\]
• When multiplying, 1 raised to any power will always remain 1. Therefore:\[1^5 = 1\]

It's crucial to handle the exponent first in any mathematical expression, especially when more operations are involved, following the order of operations.
Understanding exponentiation helps in simplifying expressions and solving equations effectively. By mastering this concept, students enhance their problem-solving skills.

Negative Sign

A negative sign in mathematical expressions can significantly alter the result. It represents a number's position on the number line, inversely flipping its value. In the expression \[-1^5\], it's important to clarify how the negative sign interacts with the exponentiation.

Here are key points to consider:

• The negative sign in \[-1^5\] applies to the whole expression, after the exponentiation is solved.
• According to the order of operations, which is Parentheses, Exponents, Multiplication/Division, Addition/Subtraction (PEMDAS), exponentiation is handled before applying the negative sign.
• Therefore, you first evaluate \[1^5 = 1\], and only then apply the negative sign, resulting in:\[-(1) = -1\]

By correctly applying negative signs, you ensure accurate results in calculations, avoiding common pitfalls.

Evaluate Expressions

Evaluating expressions like \[-1^5\] requires careful adherence to mathematical rules. The goal is to simplify and solve the expression systematically by applying the order of operations.

To evaluate effectively:

• Start with the exponentiation part of the expression, resolving the power of the given base. Here, \[1^5 = 1\].
• Next, apply any negative signs as per the preceding result. This turns the value from positive to negative: \[-(1) = -1\].

Understanding these steps ensures clarity in complex problems. It's beneficial to go through expressions slowly, ensuring each operation is followed sequentially to produce the correct outcome.

By mastering these procedures, learners can approach math expressions with greater confidence, enhancing their computational skills.