Simplify

Problem

(√(,x)+1/√(3,x))^2

Solution

1. Rewrite the radicals as fractional exponents to make the algebraic manipulation easier.

√(,x)=x^(1/2)

1/√(3,x)=x^(−1/3)

2. Apply the binomial expansion formula (a+b)^2=a^2+2*a*b+b^2 to the expression.

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

3. Simplify each term using the laws of exponents, specifically (x^a)^b=x^(a*b) and x^a⋅x^b=x^(a+b)

(x^(1/2))^2=x^1=x

2*(x^(1/2))*(x^(−1/3))=2*x^(1/2−1/3)=2*x^(3/6−2/6)=2*x^(1/6)

(x^(−1/3))^2=x^(−2/3)

4. Combine the simplified terms into the final expression.

x+2*x^(1/6)+x^(−2/3)

5. Convert the fractional exponents back into radical form if desired.

x+2√(6,x)+1/√(3,x^2)

Final Answer

(√(,x)+1/√(3,x))^2=x+2√(6,x)+1/√(3,x^2)

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