Simplify square root of cube root of 11a^7b

Problem

√(,√(3,11*a7*b))

Solution

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

√(,√(3,11*a7*b))=((11*a7*b)(1/3))(1/2)

2. Apply the power of a power rule (xm)n=x(m⋅n) to combine the exponents.

(11*a7*b)(1/3⋅1/2)=(11*a7*b)(1/6)

3. Distribute the exponent to each factor inside the parentheses using the rule (x*y*z)n=xn*yn*zn

11(1/6)*a(7/6)*b(1/6)

4. Extract the perfect sixth power from the variable a7 by rewriting it as a6⋅a

11(1/6)*(a6⋅a)(1/6)*b(1/6)

5. Simplify the term (a6)(1/6) to a and group the remaining terms back under a single radical.

a*(11*a*b)(1/6)

6. Convert the expression back into radical notation.

a√(6,11*a*b)

Final Answer

√(,√(3,11*a7*b))=a√(6,11*a*b)

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