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Chapter 0: Problem 101

Factor and simplify each algebraic expression. $$(4 x-1)^{\frac{1}{2}}-\frac{1}{3}(4 x-1)^{\frac{3}{2}}$$

Short Answer

Expert verified
\((4x -1)^{\frac{1}{2}} * (1 - \frac{4x}{3} + \frac{1}{3}) = (4x -1)^{\frac{1}{2}} (\frac{-4x+4}{3})\)

Step by step solution

01

Identifying the Common Factors

Notice that both parts of the given expression \((4x-1)^{\frac{1}{2}}\) and \(-\frac{1}{3}(4 x-1)^{\frac{3}{2}}\) have the base of \((4x -1)\) raised to different powers. This means this base is a common factor.
02

Factor the Expression

We can rewrite \((4x -1)^{\frac{3}{2}}\) as \((4x -1)^{\frac{1}{2}*3}\) which is equivalent to \((4x -1)^{\frac{1}{2}} * (4x -1)^{\frac{1}{2}*2} = (4x -1)^{\frac{1}{2}}*(4x -1)\). Therefore, the whole expression now becomes \((4x -1)^{\frac{1}{2}} - \frac{1}{3}(4x -1)^{\frac{1}{2}} * (4x -1)\), which now shows the common factor clearly.
03

Apply the Factorization

The common factor here is \((4x -1)^{\frac{1}{2}}\). We can factor the expression by putting this term in front: \((4x -1)^{\frac{1}{2}} * (1 - \frac{1}{3} * (4x -1))\).
04

Simplify the Expression

Finally, distribute the \(-\frac{1}{3}\) to get ans: \((4x -1)^{\frac{1}{2}} * (1 - \frac{4x}{3} + \frac{1}{3})\)

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Most popular questions from this chapter

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