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 a^m⋅a^n=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=3^2

(3^2)^(4/9)=3^(2⋅4/9)

3^(2⋅4/9)=3^(8/9)

6. Convert back to radical form if necessary.

3^(8/9)=√(9,3^8)

√(9,3^8)=√(9,6561)

Final Answer

√(3,9)⋅√(9,9)=3^(8/9)

Want more problems? Check here!