Problem 93 - College Algebra Essentials

Question

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

Step-by-Step Solution

Answer

The factorized and simplified form of the expression \(x^{\frac{3}{2}}-x^{\frac{1}{2}}\) is \(x^{\frac{3}{2}}\).

Step 1: Identification of the common factor

The given expression is \(x^{\frac{3}{2}}-x^{\frac{1}{2}}\). Here \(x^{\frac{1}{2}}\) is the common factor in both terms.

Step 2: Factorization of the expression

Keeping \(x^{\frac{1}{2}}\) as common factor from each term, we get \(x^{\frac{1}{2}}(x^{\frac{3}{2} - 1})\). This reduces to \(x^{\frac{1}{2}}(x) = x^{\frac{1}{2}}x\).

Step 3: Simplification of the expression

When we multiply \(x^{\frac{1}{2}}\) and \(x\), the powers add up as per the rules of power math: \(a^m × a^n = a^{m+n}\). So it becomes \(x^{\frac{3}{2}}\).