Simplify

Problem

√(4,x⋅√(3,x))/√(6,x)

Solution

1. Rewrite the radicals as fractional exponents using the rule √(n,xm)=xm/n

√(3,x)=x1/3

√(6,x)=x1/6

2. Simplify the expression inside the fourth root by applying the product rule for exponents, xa⋅xb=x(a+b)

x⋅x1/3=x(1+1/3)

x(1+1/3)=x4/3

3. Apply the fourth root to the simplified inner expression using the power of a power rule, (xa)b=x(a*b)

√(4,x4/3)=(x4/3)1/4

(x4/3)1/4=x(4/3⋅1/4)

x(4/3⋅1/4)=x1/3

4. Divide the numerator by the denominator using the quotient rule for exponents, (xa)/(xb)=x(a−b)

(x1/3)/(x1/6)=x(1/3−1/6)

5. Subtract the fractions by finding a common denominator.

1/3−1/6=2/6−1/6

2/6−1/6=1/6

6. Convert the result back into radical form.

x1/6=√(6,x)

Final Answer

√(4,x⋅√(3,x))/√(6,x)=√(6,x)

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