Problem 16 - Elementary and Intermediate Algebra

Question

Complete each solution. Simplify: $$ \begin{aligned} \left(m^{1 / 3} n^{1 / 2}\right)^{6} &=(\square)^{6}\left(n^{1 / 2}\right)^{6} \\\ &=m^\square n^{6 / 2} \\ &=m^{2} n^{3} \end{aligned} $$

Step-by-Step Solution

Answer

The simplified expression is \(m^2 n^3\).

Step 1: Understand the given expression

The expression given is \( \left(m^{1/3} n^{1/2}\right)^{6} \). You are required to simplify this expression by applying the exponent rules to each term inside the brackets to the power of 6.

Step 2: Apply the Power of a Power Rule

The rule for exponents states that \((a^m)^n = a^{m \cdot n}\). Apply this rule to each component inside the parentheses: - For \(m^{1/3}\), calculate \((m^{1/3})^6 = m^{1/3 \cdot 6}\).- For \(n^{1/2}\), calculate \((n^{1/2})^6 = n^{1/2 \cdot 6}\).

Step 3: Simplify Each Component

Perform the multiplication for each exponent to simplify further:- For \(m\), \(1/3 \cdot 6 = 6/3\), which simplifies to 2. So, \(m^{1/3 \cdot 6} = m^2\).- For \(n\), \(1/2 \cdot 6 = 6/2\), which simplifies to 3. So, \(n^{1/2 \cdot 6} = n^3\).

Step 4: Combine the Simplified Components

Use the results from Step 3 to combine into the final simplified form: \(m^2 n^3\).

Key Concepts

Power of a Power Rule

When working with exponents, the power of a power rule is a useful and powerful tool. This rule tells us how to handle situations where an exponent is raised to another exponent. Specifically, for any base \( a \) and exponents \( m \) and \( n \), the rule is stated as \((a^m)^n = a^{m \cdot n}\).

This means you multiply the exponents when you have a situation like this. In the original exercise, both \( m^{1/3} \) and \( n^{1/2} \) are inside a parenthesis and raised to the power of 6. Applying the power of a power rule means:

• For \( m^{1/3} \), we find \((m^{1/3})^6 = m^{1/3 \cdot 6} = m^2\).
• For \( n^{1/2} \), we compute \((n^{1/2})^6 = n^{1/2 \cdot 6} = n^3\).

By using this rule, you break down complex expressions into simpler terms, making computation straightforward. It's a fundamental rule that simplifies many algebraic expressions easily.

Simplifying Expressions

Simplifying expressions means reducing them to their simplest form, where no more multiplication, division, or application of exponential rules can further reduce the expression. This often involves applying various mathematical rules, including the power of a power rule, product rule, and quotient rule for exponents.

In the original exercise, simplifying involves breaking down the components inside the parentheses and then applying rules to make the expression less complex. Here’s what happens:

• You start with \(\left(m^{1/3} n^{1/2}\right)^6\).
• Apply the power of a power rule to each term inside: \( (m^{1/3})^6\) and \((n^{1/2})^6\).
• You find \(m^{2}\) and \(n^{3}\) as outputs.

Once these steps are completed, the simplified expression is \(m^2 n^3\).
By simplifying, you make algebraic expressions easier to handle, compare, and evaluate, which is crucial for solving equations efficiently.

Algebraic Expressions

Understanding algebraic expressions is key to mastering algebra. These expressions consist of variables, constants, and exponents combined through operations such as addition, subtraction, multiplication, and division. In algebra, we often encounter expressions that need to be simplified or solved for specific values.

Consider the expression from the exercise: \(\left(m^{1/3} n^{1/2}\right)^6\). This is a typical algebraic expression featuring:

• Variables \(m\) and \(n\), which represent numbers but are not specified.
• Exponents, which indicate the power each variable is raised to.
• Parentheses, denoting that everything inside should be considered one unit when applying an operation like raising to a power.

The goal is to understand how these parts interact. By breaking the expression down using rules like the power of a power rule, we make an algebraic expression much simpler. This understanding is vital, as it allows you to tackle more complicated equations in mathematics with greater ease.