Problem 50 - Algebra A Combined Function

Question

Use the properties of exponents to simplify each expression. Write with positive exponents. $$ \left(32^{1 / 5} x^{2 / 3}\right)^{3} $$

Step-by-Step Solution

Answer

The simplified expression is \(32^{3/5} \cdot x^2\).

Step 1: Apply the Power of a Power Property

The power of a power property states that when you have an exponent raised to another exponent, you multiply the exponents together. Therefore, we start by applying this property to the given expression: \(\left(32^{1/5} x^{2/3}\right)^{3} = 32^{(1/5) \cdot 3} \cdot x^{(2/3) \cdot 3}\).

Step 2: Simplify the Exponents

Next, calculate the new exponents by performing the multiplication: \(32^{(1/5) \cdot 3} = 32^{3/5}\) and \(x^{(2/3) \cdot 3} = x^{2}\). This results in the expression \(32^{3/5} \cdot x^2\).

Step 3: Express All Exponents as Positive

The expression already contains positive exponents, so no changes are necessary to express the terms positively. The final simplified expression is \(32^{3/5} \cdot x^2\).

Key Concepts

Simplifying Expressions: A Step-By-Step Guide

Simplifying expressions is a crucial skill in algebra that involves reducing expressions to their simplest form. This helps make complicated mathematical problems more manageable and easier to solve. When simplifying expressions with exponents, it's essential to apply the properties of exponents correctly. By doing so, you can transform expressions into a form that is much simpler and easier to understand.

To simplify an expression, follow these basic steps:

• Identify common rules and properties of exponents applicable to the expression.
• Apply any relevant properties to combine or reduce exponents.
• Simplify further by performing necessary arithmetic operations, such as multiplication or division, on the exponents.
• Re-write the expression using positive exponents if needed.

In the given exercise, the task was to simplify \( (32^{1/5} x^{2/3})^3 \). This involves using properties like the power of a power property to condense and express the expression in a simpler form. By doing this, we arrive at \( 32^{3/5} \cdot x^2 \), a clear and concise result.

Understanding Positive Exponents

Positive exponents signify the number of times a base number is multiplied by itself. They make expressions easier to handle and understand since they represent straightforward repeated multiplication. Negative exponents, on the other hand, indicate a reciprocal or division pattern when simplifying them, making them slightly trickier to handle.

When simplifying expressions, transforming any negative exponents into positive ones ensures a clear representation. It allows for easier computation and comparison since you deal strictly with multiplication rather than reciprocals.

In our exercise, initially, the expression \( (32^{1/5} x^{2/3})^3 \) was tackled without negative exponents. The outcome, \( 32^{3/5} \cdot x^2 \), naturally consisted of positive exponents, illustrating that each part of the expression remained a straightforward representation of repeated multiplication.

The Power of a Power Property in Exponents

The power of a power property is a fundamental concept when working with exponents. It states that when an exponential expression is raised to another power, you multiply the exponents together.

For example, if you have an expression \( (a^m)^n \), applying the power of a power property simplifies it to \( a^{m \cdot n} \). This property helps condense complex exponential expressions into a simpler form, making it easier to perform further operations like multiplication and division.

In our exercise, we applied this property to the initial expression \( (32^{1/5} x^{2/3})^3 \). By using the power of a power property, the exponents were multiplied together, resulting in \( 32^{3/5} \cdot x^{2} \). This step was crucial in transforming the original expression into a simpler, more digestible form.