Simplify

Problem

√(4,x⋅√(3,x))/√(6,x)

Solution

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

√(3,x)=x^1/3

√(6,x)=x^1/6

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

x⋅x^1/3=x^(1+1/3)

x^(1+1/3)=x^4/3

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

√(4,x^4/3)=(x^4/3)^1/4

(x^4/3)^1/4=x^(4/3⋅1/4)

x^(4/3⋅1/4)=x^1/3

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

(x^1/3)/(x^1/6)=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.

x^1/6=√(6,x)

Final Answer

√(4,x⋅√(3,x))/√(6,x)=√(6,x)

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