Simplify cube root of square root of 729x^6

Problem

√(3,√(,729*x^6))

Solution

1. Express the radicals as fractional exponents using the rule √(n,a)=a^(1/n)

((729*x^6)^(1/2))^(1/3)

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

(729*x^6)^(1/6)

3. Distribute the exponent to each factor inside the parentheses using the rule (a*b)^n=a^n*b^n

729^(1/6)⋅(x^6)^(1/6)

4. Evaluate the sixth root of the constant by identifying that 3^6=729

729^(1/6)=3

5. Simplify the variable term by multiplying the exponents.

(x^6)^(1/6)=x^(6⋅1/6)=x

6. Combine the results to find the final simplified expression.

3*x

Final Answer

√(3,√(,729*x^6))=3*x

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