Problem 81 - Intermediate Algebra

Question

Simplify each expression. Write the answers without negative exponents. All variables represent positive real numbers. See Example 8. $$ \left(m^{2 / 3} m^{1 / 3}\right)^{6} $$

Step-by-Step Solution

Answer

The simplified expression is \(m^6\).

Step 1: Combine Exponents Inside the Parentheses

First, combine the exponents of the base \(m\) inside the parentheses. When multiplying terms with the same base, you add the exponents: \[m^{2/3} imes m^{1/3} = m^{(2/3 + 1/3)}\] This simplifies to:\[m^1 = m\]

Step 2: Apply the Power of a Power Rule

Once the expression inside the parentheses is simplified to \(m\), apply the rule \((a^m)^n = a^{m imes n}\) for the entire expression:\[(m)^6 = m^{1 imes 6}\]Simplifying gives:\[m^6\]

Step 3: Simplify and Ensure No Negative Exponents Are Present

The expression \(m^6\) is already simplified and does not contain any negative exponents. Since the base \(m\) and the exponent are positive, no further simplification is needed.

Key Concepts

Exponent Rules

Exponent rules are fundamental when working with exponents, simplifying expressions, and solving equations. They provide a systematic way to manipulate expressions that involve powers. Here are a few basic exponent rules to keep in mind:

• Product of Powers Rule: When multiplying two expressions with the same base, add the exponents: \[a^m \times a^n = a^{m+n}\]
• Quotient of Powers Rule: When dividing two expressions with the same base, subtract the exponents: \[\frac{a^m}{a^n} = a^{m-n}\]
• Power of a Power Rule: When raising an expression to another power, multiply the exponents: \[(a^m)^n = a^{m \cdot n}\]

Understanding these rules is crucial for simplifying complex algebraic expressions. For example, in the given exercise, the product of powers rule helps simplify \(m^{2/3} \times m^{1/3}\) to \(m^1\). This foundational step enables further manipulation of the expression without confusion.

Positive Exponents

Positive exponents refer to the exponents that are greater than zero. They indicate how many times a base is multiplied by itself. For example, in the expression \(a^3\), the exponent 3 tells us that the base \(a\) is used three times in a multiplication: \(a \times a \times a\).
Positive exponents are easy to work with because they result in straightforward multiplication. They also ensure that the expression is kept in its simplest form without involving fractions or negative numbers. When simplifying expressions, it's important to convert any negative exponents into positive ones to follow this rule, which the final solution in our exercise already adheres to by ending with \(m^6\). Always remember, when simplifying, aim for expressions with positive exponents to maintain clarity and simplicity.

Power of a Power Rule

The power of a power rule is a key concept when working with exponents. It states that when you raise a power to another power, you should multiply the exponents. This can be expressed as: \[(a^m)^n = a^{m \cdot n}\].
Applying this rule is straightforward when you understand it involves just multiplying the two exponents involved, rather than adding or other operations.
In our exercise, after simplifying the expression inside the parentheses to \(m\), the expression \((m)^6\) applies this rule directly. By multiplying the inner exponent (which is 1) by the outer exponent (6), the expression becomes \(m^6\). This step simplifies the expression whilst ensuring no negative exponents appear. Such simplification using the power of a power rule is essential in reducing complex exponential expressions to their simplest form.