Problem 24 - Elementary and Intermediate Algebra

Question

Simplify. $$ -2^{5} $$

Step-by-Step Solution

Answer

-32

Step 1: Understand the Expression

The expression to simplify is \(-2^{5}\). Note that the negative sign is outside the exponentiation.

Step 2: Calculate the Power

Find the value of \(2^{5}\). This means multiplying 2 by itself 5 times: \(2 \times 2 \times 2 \times 2 \times 2 = 32\).

Step 3: Apply the Negative Sign

Since the negative sign is outside the exponentiation, apply it to the result from the previous step: \(-32\).

Key Concepts

Negative Exponents

Negative exponents can be tricky, but they follow a simple rule: any number raised to a negative exponent is the reciprocal of that number raised to the positive of the exponent. For example, for any nonzero number 'a' and positive integer 'n', ewline ewline 1. ewline ewline ewline ewline 2. **Understand the Placement**
This concept is key when dealing with negative signs and exponents. In the exercise, the negative sign is separate from the exponentiation. This means it doesn't follow the rule of negative exponents directly, which is why it's applied after calculating the power.
Common mistake: confusing the placement and role of the negative sign here can cause errors.

Order of Operations

Order of operations is a fundamental principle in algebra that dictates the sequence in which operations should be performed. Remembering this sequence is often made easier with the acronym PEMDAS:

• **P**arentheses
• **E**xponents
• **M**ultiplication and **D**ivision (from left to right)
• **A**ddition and **S**ubtraction (from left to right)

In the given exercise, the order of operations clarifies that the exponent should be calculated before applying the negative sign.

• First, compute the exponent: 2⁵ = 32
• Then, apply the negative sign: -32

Basic Algebra

Basic algebra revolves around understanding fundamental operations and properties. In our exercise, we deal with simplifying an expression by following the defined algebraic rules. Some important aspects include:

• **Exponents**: Understand that an exponent represents repeated multiplication of the base (e.g., 2⁵ means 2 multiplied by itself 5 times).
• **Negative Numbers**: Recognize the position and impact of negative signs in expressions. In -2⁵, the 2⁵ is evaluated first, then the negative sign is applied.
• **Simplification**: Follow the rules of algebra to simplify expressions step-by-step, ensuring no shortcuts that could lead to mistakes.

Start with the exponentiation, then move to apply the necessary signs or other operations.