Problem 100 - Intermediate Algebra

Question

In the following exercises, simplify each expression. $$ \text { (a) }(-3)^{5} \text { (b) }-3^{5} $$

Step-by-Step Solution

Answer

(a) \-243\ (b) \-243\

Step 1: Simplify \((-3)^5\)

First, recognize that \((-3)^5\)\ means we are raising -3 to the power of 5. When a negative number is raised to an odd power, the result will be negative. Calculate \[ (-3)^5 = -3 \times -3 \times -3 \times -3 \times -3 = -243 \]

Step 2: Simplify \-3^5\

Recognize that \-3^5\ means only the 3 is raised to the power of 5, and then the result is made negative. Calculate \[ 3^5 = 3 \times 3 \times 3 \times 3 \times 3 = 243 \] So, \-3^5 = -243\.

Key Concepts

Negative Exponents

Negative exponents are a way of expressing the reciprocal of a base raised to a positive exponent. In other words, when you encounter a negative exponent, flip the base and turn the exponent positive. For example, consider the expression \(a^{-n}\). This is equivalent to \(\frac{1}{a^n}\), where 'a' is the base, and 'n' is the positive exponent. Here's a quick breakdown to understand better:

• \(x^{-2} = \frac{1}{x^2}\)
• \(3^{-4} = \frac{1}{3^4}\)
• \( (-2)^{-3} = \frac{1}{(-2)^3} = -\frac{1}{8} \)

Keep in mind that negative exponents make the value smaller, turning it into a fraction. Practice transforming negative exponents into their positive, reciprocal form to get comfortable with simplifying expressions.

Order of Operations

The order of operations is crucial in solving algebraic expressions correctly. Remember the acronym PEMDAS to guide you:

• P: Parentheses first
• E: Exponents (including roots, such as square roots)
• M/D: Multiplication and Division (left to right)
• A/S: Addition and Subtraction (left to right)

When multiple operations are involved, following the correct order is key. Let's look at an example:
Simplify \(8 - (3 + 5^2) \)
Step 1: Solve inside the parentheses first: \ 3 + 5 = 8 \
Step 2: Next, apply the exponent: \ 5^2 = 25 \
Step 3: Now, solve inside the parentheses completely: \ 3 + 25 = 28 \
Step 4: Finally, perform the subtraction: \ 8 - 28 = -20 \
Following these steps ensures you get the correct result every time. Ignoring the order can lead to mistakes.

Powers of Negative Numbers

Understanding the power of negative numbers is essential because the result can vary significantly based on whether the exponent is odd or even. Let's break this down:

Negative Number Raised to an Even Power:
When a negative base is raised to an even power, the result is positive. For example, \( (-2)^4 = 2^4 = 16\). This is because multiplying an even number of negative factors results in a positive product.

Negative Number Raised to an Odd Power:
When a negative base is raised to an odd power, the result is negative. For instance, \( (-3)^5 = -243 \). Multiplying an odd number of negative factors results in a negative product.
It's also crucial to pay attention to the placement of parentheses. Take \( -3^5 \) as an example. Without parentheses, the exponent only applies to the base 3, and then the negative sign is applied afterward. So, \( 3^5 = 243 \), and adding the negative sign gives you \ -243 \. In contrast, \( (-3)^5 = -243 \) because the negative 3 is raised to the 5th power. Understanding these distinctions helps avoid confusion and ensures accurate calculations.