Evaluate cube root of 9 ninth root of 9

Problem

√(3,9)⋅√(9,9)

Solution

1. Convert the radical expressions into exponential form using the rule √(n,x)=x(1/n)

√(3,9)=9(1/3)

√(9,9)=9(1/9)

2. Apply the product rule for exponents, which states am⋅an=a(m+n)

9(1/3)⋅9(1/9)=9(1/3+1/9)

3. Find a common denominator to add the fractions in the exponent.

1/3+1/9=3/9+1/9

3/9+1/9=4/9

4. Rewrite the expression with the new exponent.

9(4/9)

5. Express the base 9 as a power of 3 to see if the expression simplifies further.

9=3

(3)(4/9)=3(2⋅4/9)

3(2⋅4/9)=3(8/9)

6. Convert back to radical form if necessary.

3(8/9)=√(9,3)

√(9,3)=√(9,6561)

Final Answer

√(3,9)⋅√(9,9)=3(8/9)

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