Problem 41 - Algebra A Combined Function

Question

Use the properties of exponents to simplify each expression. Write with positive exponents. $$ a^{2 / 3} a^{5 / 3} $$

Step-by-Step Solution

Answer

The expression simplifies to \(a^{7/3}\).

Step 1: Identify the Problem

We need to simplify the expression \(a^{2/3} a^{5/3}\) using the properties of exponents, specifically the product of powers property.

Step 2: Use the Product of Powers Property

According to the product of powers property, \(a^m \cdot a^n = a^{m+n}\). Apply this property to \(a^{2/3} \cdot a^{5/3}\), resulting in \(a^{(2/3) + (5/3)}\).

Step 3: Add the Exponents

Add the exponents \(\frac{2}{3}\) and \(\frac{5}{3}\) together. Both fractions have the same denominator, so we directly add the numerators: \(\frac{2+5}{3} = \frac{7}{3}\).

Step 4: Write the Simplified Expression

After adding the exponents, the expression simplifies to \(a^{7/3}\). All exponents are positive, so no further adjustment is needed.

Key Concepts

Product of Powers Property

Understanding the product of powers property is key to simplifying expressions with exponents. This property is applied when you are multiplying two or more expressions that have the same base.
This rule states that when you multiply powers with the same base, you simply add their exponents together. Mathematically, this is expressed as:

• \( a^m \times a^n = a^{m+n} \)

By using this property, complex expressions can be reduced to simpler forms, making calculations easier. Whether the exponents are fractions, whole numbers, or a mix, this property consistently applies.
For our example, applying the product of powers property to \( a^{2/3} \times a^{5/3} \) simplifies the expression to \( a^{(2/3) + (5/3)} \).

Exponent Addition

When dealing with exponential expressions that require simplification, adding the exponents correctly is crucial. If you have matching bases and you're multiplying them, like in our example \( a^{2/3} \times a^{5/3} \), you proceed by adding the exponents together.
In this scenario, both exponents \( \frac{2}{3} \) and \( \frac{5}{3} \) share the same denominator, making the addition straightforward. You simply add the numerators:

• \( \frac{2+5}{3} = \frac{7}{3} \)

This results in a new exponent for the base \( a \), simplifying the expression to \( a^{7/3} \).
Ensuring the bases are the same is imperative, as this rule of adding exponents only holds true under that condition.

Simplifying Expressions

Simplifying expressions with exponents involves reducing them to their most basic form without altering their value. Using the properties of exponents is vital in achieving this task efficiently.
In simplification, all rules about operations involving exponents come into play. The ultimate goal is to write the expression as compact as possible, often involving positive exponents only.

• First, apply the product of powers property as needed.
• Next, carefully add or subtract exponents, taking care with fractions or any mixed numbers.
• Ensure that all exponents are positive for the final result.

As in our case, by following these steps, \( a^{2/3} \times a^{5/3} \) was reduced to the simpler \( a^{7/3} \), with positive exponents making it more manageable for further calculations.