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

Problem

√(,√(3,11*a^7*b))

Solution

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

√(,√(3,11*a^7*b))=((11*a^7*b)^(1/3))^(1/2)

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

(11*a^7*b)^(1/3⋅1/2)=(11*a^7*b)^(1/6)

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

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

4. Extract the perfect sixth power from the variable a^7 by rewriting it as a^6⋅a

11^(1/6)*(a^6⋅a)^(1/6)*b^(1/6)

5. Simplify the term (a^6)^(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*a^7*b))=a√(6,11*a*b)

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