Victoria is finding the value of n for the equation 3/sqrt14^1/6 = 14^1/n
Victoria writes 3/sqrt 14^1/6 = (14^1/6)^3 = 14^3/6= 14^1/2 so n=2
Is Victoria correct? Justify your answer.
No, Victoria is not correct.
Let's analyze Victoria's steps:
1. Victoria writes 3/sqrt 14^1/6 = (14^1/6)^3
This step is incorrect. To simplify 3/sqrt 14^1/6, we need to rationalize the denominator. This can be done by multiplying both the numerator and the denominator by sqrt(14^1/6):
3/sqrt 14^1/6 = (3/sqrt 14^1/6) * (sqrt 14^1/6/sqrt 14^1/6) = (3*sqrt 14^1/6)/(sqrt 14^1/6 * sqrt 14^1/6) = (3*sqrt 14^1/6)/(sqrt (14^1/6 * 14^1/6)) = (3*sqrt 14^1/6)/(sqrt 14^1/3) = 3*sqrt 14^1/6/sqrt 14^1/3 = 3*sqrt 14^1/6/14^1/3
2. Victoria simplifies 3*sqrt 14^1/6/14^1/3 to 14^3/6.
This step is incorrect. The expression 3*sqrt 14^1/6/14^1/3 cannot be simplified to 14^3/6.
3. Victoria concludes that n=2.
This conclusion is incorrect because the previous steps are incorrect.
To solve the equation 3/sqrt 14^1/6 = 14^1/n, we need to equate the exponents of 14 on both sides of the equation.
The exponent of 14 on the left side is 1/6.
The exponent of 14 on the right side is 1/n.
Based on the principle that if two numbers are equal, their exponents must be equal, we have:
1/6 = 1/n
To solve for n, we cross multiply:
(n)(1/6) = 1
n/6 = 1
Multiply both sides by 6:
n = 6
Therefore, Victoria is incorrect, and the correct value of n is 6.