Problem 27 - Algebra and Trigonometry

Question

Perform the indicated operations and simplify. $$ y^{1 / 3}\left(y^{2}-1\right) $$

Step-by-Step Solution

Answer

The simplified expression is \(y^{7/3} - y^{1/3}\).

Step 1: Apply the Distributive Property

To simplify an expression like \(y^{1/3}(y^2 - 1)\), start by distributing \(y^{1/3}\) to each term inside the parentheses: \(y^{1/3} \times y^2\) and \(y^{1/3} \times -1\).

Step 2: Simplify Each Term

First, simplify \(y^{1/3} \times y^2\). When multiplying expressions with the same base, you add the exponents: \(y^{1/3} \times y^2 = y^{(1/3 + 2)} = y^{7/3}\). Next, simplify \(y^{1/3} \times -1 = -y^{1/3}\).

Step 3: Combine the Terms

After simplifying each term, combine them into a single expression: \(y^{7/3} - y^{1/3}\). This is the simplified form of the original expression.

Key Concepts

Distributive Property

The Distributive Property is a fundamental concept in algebra that allows you to multiply a single term by terms inside parentheses, helping to simplify expressions. It's like distributing or sharing one factor with every term in a group.
For example, if you have an expression like \( y^{1/3} (y^2 - 1) \), you need to multiply \( y^{1/3} \) with each of the terms inside the parentheses separately. Let's break it down:

• First, multiply \( y^{1/3} \) by \( y^2 \).
• Second, multiply \( y^{1/3} \) by \(-1\).

When you carry out these multiplication steps, you effectively make it easier to handle and combine later. The Distributive Property is often represented mathematically as \( a(b + c) = ab + ac \). So applying this in algebra helps in easing complex problems into simpler components.

Exponents

Exponents are a shorthand way of showing repeated multiplication of the same number or variable. It's important to remember a few basic rules when working with exponents, especially when they need to be combined.

When you multiply two like bases, for example, you add their exponents. This rule is incredibly helpful when simplifying expressions. Let's see how this applies with \( y^{1/3} \times y^2 \).

• Both terms have the base \( y \), so add the exponents: \( \frac{1}{3} + 2 \).
• Convert 2 to a fraction: \( \frac{6}{3} \).
• Now add the fractions: \( \frac{1}{3} + \frac{6}{3} = \frac{7}{3} \).

This results in \( y^{7/3} \). Understanding how to manipulate exponents is crucial for simplifying algebraic expressions effectively.

Combining Like Terms

Combining Like Terms is the process of simplifying an expression by merging terms with the same variables and exponents. When you have like terms, you can add or subtract them to reduce the complexity of an expression.

In our original problem, after distributing and simplifying the terms, you end up with two terms: \( y^{7/3} \) and \(-y^{1/3} \).
Although these two look similar, they are not like terms due to their different exponents. This means you cannot combine \( y^{7/3} \) with \(-y^{1/3} \) into a single term.

• To combine like terms, their base and exponents must be identical.
• If you were to combine, they do not subtract or add together due to the varying exponents.

Recognizing when terms can be combined is key for simplifying expressions and creating tidy solutions.