Evaluate ( fifth root of 9)^11

Problem

(√(5,9))11

Solution

1. Rewrite the radical expression using a fractional exponent.

√(5,9)=9(1/5)

2. Apply the power of a power rule, which states (am)n=a(m⋅n)

(9(1/5))11=9(11/5)

3. Express the base as a power of 3 to simplify further, since 9=3

(3)(11/5)=3(2⋅(11/5))

4. Multiply the exponents.

3(22/5)

5. Convert the improper fraction exponent into a mixed number to separate the whole and fractional parts.

22/5=4+2/5

6. Simplify the expression using the property a(m+n)=am⋅an

3(4+2/5)=3⋅3(2/5)

7. Calculate the integer power and convert the fractional power back to radical form.

3=81

3(2/5)=√(5,3)=√(5,9)

8. Combine the parts into the final simplified radical form.

81√(5,9)

Final Answer

(√(5,9))11=81√(5,9)

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