Problem 93 - Algebra and Trigonometry

Question

Factor and simplify each algebraic expression. $$\begin{aligned} &x^{\frac{3}{2}}-x^{\frac{1}{2}}\\\ & \end{aligned}$$

Step-by-Step Solution

Answer

The simplified expression is \(x^{\frac{1}{2}}(x-\frac{1}{\sqrt{x}})\)

Step 1: Identify the common factor

The common factor between the two terms \(x^{\frac{3}{2}}\) and \(x^{\frac{1}{2}}\) is \(x^{\frac{1}{2}}\). This is because this is the smallest power between the two terms.

Step 2: Factor out the common factor

Factor out the common factor \(x^{\frac{1}{2}}\) from each term: \(x^{\frac{1}{2}}(x-x^{-\frac{1}{2}})\). To do this, subtract \(x^{\frac{1}{2}}\) from each exponent.

Step 3: Simplify the expression

The term \(x^{-\frac{1}{2}}\) is equivalent to \(\frac{1}{\sqrt{x}}\). So the expression becomes \(x^{\frac{1}{2}}(x-\frac{1}{\sqrt{x}})\).

Key Concepts

Factoring

Factoring is a key concept in algebra that involves breaking down complex expressions into simpler, more manageable parts. Essentially, when we factor an expression, we are looking for terms that are common among the parts of our algebraic expression. In the problem at hand, we've identified that the smallest power of the variable, or the base, shared by both terms is \(x^{\frac{1}{2}}\). When we factor, we take this common factor out of each term.

Here's a straightforward way to think about it:

• Identify the smallest exponent among the terms as your common factor. In this example, it was \(x^{\frac{1}{2}}\).
• "Pull out" this factor from the expression, leaving what's left as the inside terms. This transformation reduces the expression into something easier to process.

Factoring is like finding the "building blocks" of an equation. It makes complex algebra much easier and is a helpful step in solving equations.

Exponents

Understanding exponents is crucial when dealing with algebraic expressions. Exponents tell us how many times a base is multiplied by itself. In the case of fractional exponents, like \(x^{\frac{3}{2}}\) and \(x^{\frac{1}{2}}\), they introduce a bit of a twist by indicating both root and power.

Here's a breakdown:

• An expression like \(x^{\frac{3}{2}}\) means \(x^3\) followed by the square root.
• Similarly, \(x^{\frac{1}{2}}\) is another way to write the square root of \(x\), \(\sqrt{x}\).
• Understanding this notation helps simplify expressions involving roots and powers.

Exponents, even when fractional, are just tools to rewrite expressions in their various forms. They make it easier to express ideas that would otherwise be cumbersome, such as roots and powers combined.

Simplifying Expressions

Simplifying algebraic expressions involves reducing them to their simplest, most concise form. After factoring, the next step is often to simplify what's left behind. This might involve combining like terms or rewriting parts of the expression using different but equivalent terms.

In the exercise solution, after factoring out \(x^{\frac{1}{2}}\), we were left with the term \(x^{-\frac{1}{2}}\). Simplification here means transforming this negative exponent into a more familiar form with positive indices.

• The negative exponent \(x^{-\frac{1}{2}}\) can be rewritten as \(\frac{1}{\sqrt{x}}\), since negative exponents indicate reciprocals.
• The entire expression then becomes \(x^{\frac{1}{2}}(x - \frac{1}{\sqrt{x}})\).

Simplifying is about finding the clearest way to represent an expression, which can often reveal solutions or make further calculations easier. It's like tidying up an algebraic "sentence" to ensure it's understood correctly.