John claims that 3^2/3 * 3^3 to the 6th root = 3^2/3 * 3^3/6 = 3^6/18 = cubed root of 3
Is he correct? Explain why not justify your answer and show your work
No, John is not correct.
When multiplying powers with the same base, you add the exponents. So, 3^(2/3) * 3^(3/6) would be equal to 3^(2/3 + 3/6).
Let's simplify the exponents:
2/3 + 3/6 = 2/3 + 1/2 = (4/6) + (3/6) = 7/6.
Therefore, 3^(2/3) * 3^(3/6) = 3^(7/6).
To convert this into a simplified fractional exponent, we can rewrite 3^(7/6) as the 6th root of 3^7 raised to the power of 2.
So, 3^(7/6) = (6th root of 3^7)^2 = (cubed root of 3)^2 = 3^(2/1) = 3^2 = 9.
Therefore, the correct simplification is 3^(2/3) * 3^(3/6) = 9, not the cubed root of 3.