Victoria is finding the value of n for the equation
3^√14^1/6 = 14^1/n
Victoria writes
3√14³ = (146) ³ = 14³ = 14 so
n = 2
Is Victoria correct? Justify your answer.
No, Victoria is incorrect.
In her solution, she raised both sides of the equation to the power of 3, which is not a valid step. In general, when raising an equation to a power, you must raise both sides of the equation to that same power.
Let's solve the equation correctly:
Let's start by simplifying both sides of the equation individually:
On the left side, we have 3^(√14^(1/6)).
Since the exponent (√14^(1/6)) is inside the root, we can rewrite the expression as (14^(1/6))^(1/2).
And since we have two exponents being multiplied, we multiply them: (14^(1/6))^(1/2) = 14^((1/6) * (1/2)).
This simplifies to 14^(1/12).
On the right side, we have 14^(1/n).
Setting the simplified left side equal to the right side, we have:
14^(1/12) = 14^(1/n).
Since both sides of the equation have the same base (14), in order for the two expressions to be equal, the exponents must be equal:
1/12 = 1/n.
To find the value of n, we can cross-multiply:
1*n = 12*1.
This simplifies to n = 12.
Therefore, the correct value of n is 12, not 2.