Problem 58 - Elementary and Intermediate Algebra

Question

Simplify. Do not use negative exponents in the answer. \(m^{10} \cdot m^{-6}\)

Step-by-Step Solution

Answer

The expression simplifies to \(m^4\).

Step 1: Identify the Problem

We need to simplify the expression given: \(m^{10} \cdot m^{-6}\). This involves the operation of exponents where we multiply terms with the same base.

Step 2: Apply the Exponent Multiplication Rule

When multiplying terms with the same base, you add the exponents. So, we have \(m^{10} \cdot m^{-6} = m^{10 + (-6)}\).

Step 3: Simplify the Exponents

Calculate the result of adding the exponents: \(10 + (-6) = 4\). Thus, the expression simplifies to \(m^4\).

Step 4: Write the Final Answer

Since exponents are positive, \(m^4\) is the final simplified version of the expression with no negative exponents.

Key Concepts

Exponent Rules

Understanding the rules of exponents is crucial for simplifying expressions in algebra. Exponent rules allow us to manipulate powers of numbers or variables following specific guidelines. Here are some key rules

• Multiplication Rule: When multiplying two powers with the same base, you add the exponents. For example, if you have \(a^m \cdot a^n\), it simplifies to \(a^{m+n}\).
• Division Rule: When dividing two powers with the same base, you subtract the exponents, as shown by \(a^m / a^n = a^{m-n}\).
• Power Rule: Raising a power to another power involves multiplying the exponents: \((a^m)^n = a^{m \cdot n}\).
• Zero Exponent Rule: Any number with an exponent of zero equals one: \(a^0 = 1\) (provided \(aeq0\)).

Applying these rules helps us simplify complex expressions, like \(m^{10} \cdot m^{-6}\), making it easier to solve algebraic problems.

Multiplication of Exponents

When multiplying exponents, especially those with the same base, our focus is on adding the exponents together. This simplifies expressions and reduces complexity. Let's break this down.Consider the expression \(m^{10} \cdot m^{-6}\). Both terms have the base \(m\). According to the multiplication of exponents rule, add the exponents:

• Calculate \(10 + (-6) = 4\).
• The expression becomes \(m^4\). This result gives us a single term, which is simpler to understand and work with.

This straightforward approach underpins many operations in algebra, enabling clear and efficient simplification of expressions. It is especially useful in handling variables and ensuring that we follow streamlined processes every time.

Positive Exponents

The concept of positive exponents is integral in the simplification of expressions. It means that all exponents in the expression should be greater than or equal to zero. Positive exponents are straightforward as they represent the standard repeated multiplication of the base.In cases where simplifying results in negative exponents, it's important to rewrite them. Consider a scenario where you simplify an expression and end up with \(m^{-2}\). In keeping with the rule of only positive exponents, you would rewrite it as \(1/m^2\), converting the negative exponent to a positive one by moving it to the denominator.In our worked example, \(m^{10} \cdot m^{-6}\) simplifies directly to \(m^4\), which already has a positive exponent. Ensuring all results use only positive exponents helps maintain clarity and prevents errors in calculations. This practice is crucial in more advanced algebraic manipulations where expressions can get even more complex.