Problem 84 - Elementary and Intermediate Algebra

Question

Simplify each expression. Write the answers without negative exponents. All variables represent positive real numbers. See Example 8. $$ \left(m n^{-2 / 3}\right)^{-3 / 5} $$

Step-by-Step Solution

Answer

The expression simplifies to \(\frac{n^{2/5}}{m^{3/5}}\).

Step 1: Understanding Negative Exponents

The expression \((mn^{-2/3})^{-3/5}\) involves negative exponents. Recall that a negative exponent \(n^{-a}\) is equivalent to \(\frac{1}{n^a}\). This property will be used to eliminate negative exponents in subsequent steps.

Step 2: Apply Power of a Product Rule

The given expression can be rewritten as \((m^{-3/5})(n^{2/3 \times -3/5})\). This involves distributing the outer exponent to both bases inside the parentheses. Use the property \((ab)^c = a^c \cdot b^c\) to separate the terms.

Step 3: Simplify Exponents

Now find the exponent for each component: \(m^{1 \cdot -3/5} = m^{-3/5}\) and \(n^{(-2/3) \cdot (-3/5)} = n^{2/5}\). The expression becomes \(m^{-3/5} \cdot n^{2/5}\).

Step 4: Rewrite Without Negative Exponents

Rewrite \(m^{-3/5}\) as \(\frac{1}{m^{3/5}}\). The full expression becomes \(\frac{n^{2/5}}{m^{3/5}}\). This removes the negative exponent by converting it into a reciprocal.

Key Concepts

Negative Exponents

Negative exponents can seem a bit confusing at first, but they're simply a way to represent the reciprocal of a number raised to a positive exponent. When you see a negative exponent, like in the expression \(n^{-a}\), it means you need to take the reciprocal of \(n^a\). We can express this concept as \(n^{-a} = \frac{1}{n^a}\). This is handy when you're trying to simplify expressions, especially when you're aiming to get rid of negative exponents entirely.

• Turning negative exponents positive: Use the reciprocal.
• Keep all variables positive to simplify further operations.

In the exercise, notice how simplifying \(mn^{-2/3}\) required using this property, which later became crucial for rewriting it without negative exponents.

Power of a Product Rule

The Power of a Product Rule is a fundamental concept in algebra that helps to simplify expressions involving exponents. This rule states that when you have a product raised to an exponent, you can apply the exponent to each part of the product separately. It's expressed as \((ab)^c = a^c \cdot b^c\).

• Distribute an outer exponent to all terms inside the parentheses.
• Handle each term independently for simplification.

In the original problem, the expression \((mn^{-2/3})^{-3/5}\) needed simplification using this rule. Each term inside the parentheses was raised separately, resulting in \(m^{-3/5}\) and \(n^{2/5}\). This simplification makes expressions easier to work with, especially when you're simplifying further or eliminating negative exponents.

Simplifying Expressions

Simplifying algebraic expressions is a method used to make expressions more manageable and easier to interpret. This involves reducing expressions to their simplest form by performing a series of mathematical operations.

• Use exponent rules to combine or reduce powers.
• Transform complex expressions into simpler, more understandable forms.

The given expression \(m^{-3/5} \cdot n^{2/5}\) is simplified through the conversion of negative exponents into positive ones. Here, \(m^{-3/5}\) was rewritten as \(\frac{1}{m^{3/5}}\), while \(n^{2/5}\) remained as is because its exponent was already positive. The final simplified expression became \(\frac{n^{2/5}}{m^{3/5}}\), free of negative exponents and ready for any further operations or evaluations. This simplification ensures clarity and ease in handling algebraic tasks.