Problem 30 - College Algebra

Question

For the following exercises, simplify the given expression. Write answers with positive exponents. $$ \left(w^{0} x^{5}\right)^{-1} $$

Step-by-Step Solution

Answer

The expression simplifies to \( \frac{1}{x^5} \).

Step 1: Understand the Expression

The expression given is \( \left(w^0 x^5\right)^{-1} \). It includes a power of zero, a positive exponent, and the negative exponent outside the parentheses.

Step 2: Simplify Inside the Parentheses

Recall that any non-zero number to the power of zero is 1. Thus, \( w^0 = 1 \). Hence, the expression inside the parentheses becomes \( 1 \cdot x^5 = x^5 \). So, \( \left(w^0 x^5\right) \) simplifies to \( x^5 \).

Step 3: Apply Negative Exponent Rule

The expression \( \left(x^5\right)^{-1} \) indicates that we need to apply the negative exponent rule, which states \( a^{-n} = \frac{1}{a^n} \). Therefore, \( \left(x^5\right)^{-1} = \frac{1}{x^5} \).

Step 4: Final Simplified Expression with Positive Exponents

The entire expression simplifies to \( \frac{1}{x^5} \) as per the rules of exponents. All exponents are positive in the final result.

Key Concepts

Negative Exponents

Dealing with negative exponents can be confusing at first, but they have a simple and logical rule. When you see a negative exponent, it signifies the inverse or reciprocal of the base number raised to the corresponding positive exponent. For example, the expression \( a^{-n} \) can be rewritten as the reciprocal \( \frac{1}{a^n} \). This means you take 1 and divide it by the base raised to the positive version of the exponent.

In the context of our exercise, \( (x^5)^{-1} \) illustrates this principle. The base is \( x \), and the exponent \(-1\) indicates we need the reciprocal. Thus, it translates to \( \frac{1}{x^5} \). Getting comfortable with changing negative exponents to positive by flipping the fraction is crucial in simplifying expressions.

Positive Exponents

Positive exponents indicate how many times to multiply the base by itself. For instance, \( x^5 \) means \( x \) is multiplied by itself five times: \( x \times x \times x \times x \times x \). These are the basic and generally straightforward kind of exponents. They help in expanding and simplifying expressions neatly.

When simplifying expressions, it's ideal to have all exponents in their positive state, as it makes the calculations cleaner and more straightforward. In solving the exercise, the final expression required positive exponents, transforming \( (x^5)^{-1} \) into \( \frac{1}{x^5} \), assuring a simpler and standardized form.

Exponent Rules

Exponent rules are vital for simplifying expressions accurately. These rules include several key principles:

• Power of Zero: Any number raised to the power of zero, except zero itself, is 1. Thus, \( a^0 = 1 \).
• Negative Exponent Rule: Converts expressions with negative exponents to positive by taking the reciprocal: \( a^{-n} = \frac{1}{a^n} \).
• Multiplication Rule: To multiply like bases, add the exponents: \( a^m \times a^n = a^{m+n} \).
• Division Rule: To divide like bases, subtract the exponents: \( \frac{a^m}{a^n} = a^{m-n} \).

In the exercise provided, the key rules applied were the power of zero and negative exponent rule. These helped in transforming and simplifying \( \left(w^0 x^5\right)^{-1} \) to a form with positive exponents, demonstrating the power and utility of these exponent principles.