Problem 44 - Intermediate Algebra

Question

Use the properties of exponents to simplify each expression. Write with positive exponents. $$ y^{4 / 3} \cdot y^{-1 / 3} $$

Step-by-Step Solution

Answer

The simplified expression is \( y \).

Step 1: Identify the Base and Exponents

The expression given is \( y^{4/3} \cdot y^{-1/3} \), where both terms have the same base \( y \). The exponents are \( \frac{4}{3} \) and \( -\frac{1}{3} \) respectively.

Step 2: Use the Product of Powers Property

According to the product of powers property, if you multiply like bases \( a^m \cdot a^n \), you add the exponents: \( a^{m+n} \). Apply this property to \( y^{4/3} \cdot y^{-1/3} \): \[y^{4/3} \cdot y^{-1/3} = y^{(4/3) + (-1/3)}\]

Step 3: Simplify the Exponent

Add the exponents \( \frac{4}{3} \) and \( -\frac{1}{3} \):\[\frac{4}{3} + (-\frac{1}{3}) = \frac{4}{3} - \frac{1}{3} = \frac{3}{3} = 1\]

Step 4: Simplify the Expression

Rewrite the expression with the simplified exponent:\[y^{4/3} \cdot y^{-1/3} = y^1\]Which simplifies to \( y \) because anything raised to the power of 1 is the base itself.

Key Concepts

Product of Powers Property

When dealing with exponents, one of the most efficient rules is the product of powers property. This property helps you simplify expressions where you multiply numbers or variables with the same base. Let's break it down:

• If you have two expressions with the same base, such as \(a^m \cdot a^n\), you can combine them into a single expression by adding the exponents together.
• Mathematically, this is expressed as \(a^m \cdot a^n = a^{m+n}\).

It's important to only add the exponents when the bases are identical. In our exercise, we used this property to combine \(y^{4/3}\) and \(y^{-1/3}\) because they share the base \(y\). The rule makes it easier to simplify complex exponential expressions quickly and is especially useful when dealing with fractional and negative exponents.

Simplifying Exponents

Simplifying exponents is a fundamental skill in algebra that helps make expressions more manageable. It involves a few key steps that ensure your expressions are written as simply and neatly as possible:

• First, apply any relevant rules, such as the product of powers property as we discussed earlier.
• Next, make sure to simplify the exponents themselves. This means performing any arithmetic necessary, such as addition or subtraction of fractions.

In our solved exercise, we simplified the exponents by calculating \(4/3 + (-1/3)\), which equals \(3/3\) or \(1\). After simplifying, the expression becomes much easier to understand: \(y^1\). This not only clarifies the expression but also reduces the chance of making errors in subsequent calculations. Always aim to express the simplified result using positive exponents to maintain clarity and standard form.

Like Bases

Exponents are much easier to manipulate when they involve like bases, which means the numbers or variables being raised to a power are the same. This commonality is crucial when applying properties such as the product of powers.Consider this scenario:

• In the expression \(y^{4/3} \cdot y^{-1/3}\), the like base is \(y\). Both terms are centered around \(y\), allowing you to merge the two terms with the product of powers property.
• If the bases were different, such as \(x^{4/3} \cdot y^{-1/3}\), this property would not apply, and additional methods would be needed to simplify the expression.

Recognizing like bases is the first step in deciding which property to use, making it a foundational concept when working with exponent rules. By understanding and identifying like bases, you can apply exponential properties correctly and simplify your work efficiently.