Problem 30 - College Algebra with Corequisite Support

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 simplified expression is \(\frac{1}{x^5}\).

Step 1: Apply the Zero Exponent Rule

According to the zero exponent rule, any base raised to the power of zero equals 1. So, simplify the expression using this rule: \[w^0 = 1\] Thus, the expression simplifies to: \[(1 \cdot x^5)^{-1}\] which is \[x^5^{-1}\].

Step 2: Apply the Negative Exponent Rule

The negative exponent rule states that \(a^{-m} = \frac{1}{a^m}\). Apply this rule to the expression \(x^5^{-1}\):\[x^5^{-1} = \frac{1}{x^5}\].

Step 3: Final Simplified Expression

After converting the original expression using the zero exponent and negative exponent rules, the fully simplified form is:\[\frac{1}{x^5}\].

Key Concepts

Zero Exponent Rule

The zero exponent rule tells us that any number or variable raised to the power of zero equals 1. This might seem surprising at first. How can, say, \(x^0\) be 1? Think about it this way: as you reduce the exponent of a number or variable by one, you are essentially dividing by the base. When you reach an exponent of zero, this means you've divided the base by itself, resulting in 1.

• For any non-zero number \(a\), \(a^0 = 1\).
• This rule is crucial for simplifying expressions easily, especially when simplifying expressions with variables.

In the given problem, applying the zero exponent rule simplifies \(w^0\) to 1, which helps in simplifying the entire expression step-by-step.

Negative Exponent Rule

Negative exponents can be puzzling, but they serve a useful purpose. The negative exponent rule implies that a number with a negative exponent represents the reciprocal of the number with a positive exponent.

• If you encounter \(a^{-m}\), it can be transformed into \(\frac{1}{a^m}\).
• This transformation allows you to eliminate negative exponents from your expressions, leaving only positive ones.

For example, in our original problem, the expression \((x^5)^{-1}\) was converted to \(\frac{1}{x^5}\) by using this rule. Understanding this rule allows you to rewrite expressions in a form that is often simpler to evaluate and understand.

Positive Exponents

Positive exponents denote straightforward multiplication of a number or variable by itself. For example, \(x^5\) means \(x\) multiplied by itself five times: \(x \times x \times x \times x \times x\).

• Positive exponents essentially represent the base number being multiplied repeatedly.
• This concept helps in rewriting expressions to a more intuitive format, especially after handling zero and negative exponents.

In our problem, once the zero and negative exponents were addressed, the expression ultimately expressed the positive exponent form: \(\frac{1}{x^5}\). This outcome affirms the importance of understanding all exponent rules when simplifying expressions.