Simplify

Problem

√(,√(,b))/√(3,a*b)

Solution

1. Convert radicals to fractional exponents using the rule √(n,x)=x^(1/n)

√(,√(,b))=(b^(1/2))^(1/2)

√(3,a*b)=(a*b)^(1/3)

2. Apply the power of a power rule (x^m)^n=x^(m⋅n) to the numerator.

(b^(1/2))^(1/2)=b^(1/4)

3. Apply the power of a product rule (x*y)^n=x^n*y^n to the denominator.

(a*b)^(1/3)=a^(1/3)*b^(1/3)

4. Combine the terms into a single fraction.

(b^(1/4))/(a^(1/3)*b^(1/3))

5. Apply the quotient rule (b^m)/(b^n)=b^(m−n) to the variable b

b^(1/4−1/3)=b^(3/12−4/12)

b^(−1/12)

6. Rewrite with positive exponents using the rule x^(−n)=1/(x^n)

1/(a^(1/3)*b^(1/12))

Final Answer

√(,√(,b))/√(3,a*b)=1/(a^(1/3)*b^(1/12))

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