Problem 52 - Precalculus Mathematics for Calculus

Question

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

Step-by-Step Solution

Answer

The simplified expression is \(2x - x^{1/2}\).

Step 1: Distribute the Term

Use the distributive property to multiply the term \(x^{1/4}\) with each term inside the parentheses. Begin by distributing \(x^{1/4}\) to \(2x^{3/4}\) and then to \(-x^{1/4}\).

Step 2: Multiply Exponents

When multiplying terms with the same base, add the exponents. For \(x^{1/4} \cdot 2x^{3/4}\), this becomes \(2x^{1/4 + 3/4}\) which simplifies to \(2x^{1}\). For \(-x^{1/4} \cdot x^{1/4}\), the result is \(-x^{1/4 + 1/4}\) or \(-x^{1/2}\).

Step 3: Simplify the Expression

Combine the results from Step 2. The expression simplifies to \(2x^1 - x^{1/2}\). Write it in its simplest form, resulting in \(2x - x^{1/2}\).

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra that allows you to simplify expressions involving multiplication over addition or subtraction. It states that for any numbers or expressions, the equation \(a(b + c) = ab + ac\) holds true. Essentially, you "distribute" the term outside the parentheses to each term inside.

In the given exercise, we applied the distributive property to the expression \(x^{1/4}(2x^{3/4} - x^{1/4})\). We treated \(x^{1/4}\) as a single term that needs to be multiplied by both terms inside the parentheses: \(2x^{3/4}\) and \(-x^{1/4}\). By multiplying \(x^{1/4}\) with each of these terms individually, we separate the expression into smaller, more manageable parts.

This property helps in breaking down complex expressions and is extremely useful in algebra for simplifying calculations and solving equations.

Simplifying Expressions

Simplifying expressions involves breaking down a complex mathematical expression into its simplest form. This often means combining like terms, applying properties of operations, and reducing the expression as much as possible.

In our scenario, after we distributed \(x^{1/4}\), we got two terms: \(2x^{1}\) and \(-x^{1/2}\). To reach the final simplified expression, each term was rewritten in its simplest possible form.

• Combine constants and like terms.
• Ensure no further operations or factors can be performed to simplify the expression further.

By simplifying, the expression becomes more concise and easier to understand or use in further calculations.

In practical terms, simplifying helps us to not only make computations easier but also to interpret and understand the results correctly.

Multiplying Exponents

Exponent rules are crucial for simplifying expressions that contain powers. When multiplying terms with the same base, one of the key rules is to add the exponents. In other words, for any base \(x\), the expression \(x^a \, \cdot \, x^b = x^{a+b}\).

In the example exercise, we applied this rule during the distributive step. For instance:

• When multiplying \(x^{1/4}\) by \(2x^{3/4}\), we added the exponents: \(1/4 + 3/4 = 1\), giving us \(2x^1\).
• Similarly, multiplying \(x^{1/4}\) by \(-x^{1/4}\) involved adding the exponents: \(1/4 + 1/4 = 1/2\), resulting in \(-x^{1/2}\).

These rules make it straightforward to handle products involving powers, and they're particularly powerful tools in algebra because they allow us to quickly rewrite complex terms in a more unified and simplified form.