Problem 132 - Elementary Algebra

Question

For the following problems, write each expression so that only positive exponents appear. $$ \left(m^{0}\right)^{-1}, \quad m \neq 0 $$

Step-by-Step Solution

Answer

Question: Rewrite the expression \(\left(m^{0}\right)^{-1}\), with \(m \neq 0\), using only positive exponents. Answer: The expression \(\left(m^{0}\right)^{-1}\) is equal to \(1\).

Step 1: Apply the exponent rule for powers raised to 0

According to the rule, any nonzero number raised to the power of 0 is 1: $$ \left(m^{0}\right)^{-1} = \left(1\right)^{-1} $$

Step 2: Apply the exponent rule for negative exponents

To rewrite the negative exponent as a positive exponent, we find the reciprocal of the expression: $$ \left(1\right)^{-1} = \frac{1}{1^1} = \frac{1}{1} $$

Step 3: Simplify the expression

The simplified expression is: $$ \frac{1}{1} = 1 $$ The expression \(\left(m^{0}\right)^{-1}\), with \(m \neq 0\), rewritten with only positive exponents, is equal to \(1\).

Key Concepts

Understanding Negative Exponents

Negative exponents can often seem confusing, but they actually have a simple rule associated with them. When we see a negative exponent, it indicates a reciprocal. To solve expressions with negative exponents:

• Rewrite the base with the positive exponent by finding its reciprocal. For example, if you have \(an^{-b}\), it becomes \(\frac{1}{a^b}\).

This step is crucial in converting all exponents to positive, making any mathematical expression easier to understand and simplify. In our exercise, we used this principle to change \(1^{-1}\) into \(\frac{1}{1}\), which is simply 1. Remember:

• Negative exponents do not mean negative numbers.
• They're simply an indication of taking the reciprocal.

As you work with more complex equations, keeping this basic rule in mind will help in further calculations.

Key Exponent Rules

Exponent rules provide a foundation for working with powers and simplifying expressions efficiently. Here are some key exponent rules:

• Any number raised to the power of zero is always 1, provided the number itself is not zero. This is called the Zero Exponent Rule, as seen in \(m^0 = 1\).
• The Negative Exponent Rule states that \(a^{-n} = \frac{1}{a^n}\). It dictates that negative exponents represent reciprocal powers.
• The Power of a Power Rule says \((a^m)^n = a^{m \times n}\), simplifying expressions where numbers or variables are raised to multiple powers.

By mastering these rules, you can tackle a variety of mathematical expressions more easily. In our example, these rules helped us to transform and simplify \(\left(m^{0}\right)^{-1}\) into 1.

Simplifying Expressions with Exponents

Simplifying expressions involves applying exponent rules and arithmetic to make an expression easier to understand. Here's a simplified approach to this process:

• Identify and use the Zero Exponent Rule for any elements in the expression that involve a zero exponent, simplifying them to 1 immediately.
• Convert all negative exponents to positive by using their respective reciprocals. This helps in dealing with fractions or simplifying them further.
• Perform arithmetic operations like multiplying or dividing where necessary to combine like terms.

By following these steps, the expression \(\left(m^{0}\right)^{-1}\) became \(\frac{1}{1} = 1\). Simplifying like this removes complex elements and presents the simplest form of the original statement, making it much easier to work with.