Problem 28 - Algebra and Trigonometry

Question

Perform the indicated operations and simplify. $$ y^{1 / 4}\left(y^{1 / 2}+2 y^{3 / 4}\right) $$

Step-by-Step Solution

Answer

The simplified expression is \(y^{3/4} + 2y\).

Step 1: Distribute the terms

Start by distributing the term outside the parentheses, which is \(y^{1/4}\), to each term inside the parentheses. This results in two separate multiplication operations: \(y^{1/4} \times y^{1/2}\) and \(y^{1/4} \times 2y^{3/4}\).

Step 2: Simplify the exponents of like bases

Use the property of exponents \(a^m \times a^n = a^{m+n}\) to simplify each term:- For the first term, \(y^{1/4} \times y^{1/2}\), add the exponents: \(\frac{1}{4} + \frac{1}{2} = \frac{1}{4} + \frac{2}{4} = \frac{3}{4}\). So the term becomes \(y^{3/4}\).- For the second term, \(y^{1/4} \times 2y^{3/4}\), distribute the \(2\) as a coefficient, and add the exponents: \(\frac{1}{4} + \frac{3}{4} = \frac{4}{4} = 1\). Thus, the term becomes \(2y^1 = 2y\).

Step 3: Combine the simplified terms

Now that each term is simplified, combine them:- The expression \(y^{3/4} + 2y\) is the simplified form of the original expression \(y^{1/4}(y^{1/2} + 2y^{3/4})\).

Key Concepts

Exponent Rules

When working with algebraic expressions, exponent rules are incredibly helpful. They allow us to simplify complex terms by managing powers of the same base. A fundamental rule of exponents is the product of powers rule, which states that when multiplying like bases, you add the exponents: \(a^m \times a^n = a^{m+n}\). This rule played a major role in the given exercise.
For example, in the expression \(y^{1/4} \times y^{1/2}\), we apply this rule. Here, the base is \(y\) and the exponents \(1/4\) and \(1/2\) are added together, resulting in \(y^{(1/4 + 1/2)}\) which simplifies to \(y^{3/4}\). This simplifies the calculation by reducing the number of operations needed.
Remembering and applying these exponent rules correctly can transform algebraic expressions and make them much easier to work with and understand. Always pay attention to the bases; this rule can only be used when they are the same.

Polynomial Simplification

Simplifying polynomials involves reducing the expression to its simplest form, which means fewer terms and less complexity. This process often involves combining like terms and fully utilizing the power of exponent rules.
In the given exercise, after applying exponent rules, our polynomial became \(y^{3/4} + 2y\). This expression is already simplified as it consists of two distinct terms that cannot be combined further. Each term has the same base \(y\) but different exponents, so they remain separate.
When simplifying polynomials, ensure that you:

• Identify and combine like terms, which have the same degree and variable.
• Apply exponent rules to simplify terms where possible.
• Look for opportunities to factorize expressions or apply other algebraic techniques for simplification.

This process of simplification is crucial for solving equations efficiently and understanding the relationships within an expression.

Distributive Property

The distributive property is a key concept in algebra that allows us to simplify expressions and solve equations more effectively. It states that \(a(b + c) = ab + ac\). This means you need to multiply the term outside the parenthesis by each term inside, then sum up the results.
In our example, we applied the distributive property with \(y^{1/4}\) outside the parenthesis and \(y^{1/2} + 2y^{3/4}\) inside. This resulted in the terms \(y^{1/4} \times y^{1/2}\) and \(y^{1/4} \times 2y^{3/4}\).
Following the distributive property ensures that every part of the expression is accounted for and accurately calculated. This step is essential before moving on to simplify using exponent rules. The distributive property helps break down large problems into manageable pieces, making complex algebraic problems more approachable.
Embrace this property as it plays a huge role not only in simplification but also in equation solving and algebraic manipulation.