3^√14^1/6 = 14^1/n
Victoria writes
3√14³ = (14^1/6) ³ = 14³ = 14 so
n = 2
Is Victoria correct? Justify your answer.
No, Victoria is not correct.
To solve the equation, we need to use the properties of exponents. We know that (a^m)^n = a^(m*n).
Starting from the left side of the equation:
3^√14^1/6 = (3^√14)^(1/6) = (3^√14)^(1*1/6) = (3^√14)^(1/6) = 3^(1/6) * (√14)^(1/6)
Now, let's simplify the right side of the equation:
14^1/n = 14^(1/2)
Comparing the two sides, we have (3^√14)^(1/6) = 3^(1/6) * (√14)^(1/6) and 14^(1/2) = 3^(1/6) * (√14)^(1/6)
Since the two sides are equal, we can conclude that n = 6.
Therefore, Victoria's conclusion that n = 2 is incorrect. The correct value for n is 6.