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=a2+2*a*b+b2 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 (xa)b=x(a*b) and xa⋅xb=x(a+b)

(x(1/2))2=x1=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,x2)

Final Answer

(√(,x)+1/√(3,x))2=x+2√(6,x)+1/√(3,x2)

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