Factor x^(-3/2)+2x^(-1/2)+x^(1/2)

Problem

x^(−3/2)+2*x^(−1/2)+x^(1/2)

Solution

1. Identify the term with the lowest exponent to factor out from the expression.

2. Factor out x^(−3/2) from each term by subtracting −3/2 from each exponent.

3. Simplify the exponents inside the parentheses.

x^(−3/2)*(x^0+2*x^1+x^2)

4. Rewrite the expression inside the parentheses in standard quadratic form.

x^(−3/2)*(x^2+2*x+1)

5. Recognize that the quadratic x^2+2*x+1 is a perfect square trinomial.

6. Factor the perfect square trinomial into (x+1)^2

Final Answer

x^(−3/2)+2*x^(−1/2)+x^(1/2)=x^(−3/2)*(x+1)^2

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