Problem 96 - Algebra and Trigonometry

Question

Factor and simplify each algebraic expression. $$12 x^{-\frac{3}{4}}+6 x^{\frac{1}{4}}$$

Step-by-Step Solution

Answer

\[6x^{-\frac{3}{4}}(2+x)\]

Step 1: Identifying common factors

In the expression \(12 x^{-\frac{3}{4}}+6 x^{\frac{1}{4}}\), we start by identifying the common factors which include the numeric part and the variable part. The common numeric factor is 6 and the common variable factor is \(x^{-\frac{3}{4}}\) which is smallest power of x.

Step 2: Factor out the common factors

Now, factor out the common factors from each term of the expression which gives us: \[6x^{-\frac{3}{4}}(2+x)\]

Step 3: Simplify

We already factored out the greatest common factor in the previous step. So, there is no simplification required in this case. Therefore, the expression remains the same: \[6x^{-\frac{3}{4}}(2+x)\]