The value of (₆√27-√27/4)^2 is:
56
I don't know what you mean by
(₆
I thought at first you meant the numbers were in base 6, but then we couldn't have a 7.
I will read it as
(√27-√27/4)^2
= ( (4√27 - √27)/4 )^2
= ((3√27)/4)^2
= 9(27)/16 = 243/16
To find the value of the expression (₆√27 - √27/4)^2, we should begin by simplifying the square roots:
The sixth root of 27 can be written as (27)^(1/6), and the square root of 27/4 can be written as ((27/4)^(1/2)).
Let's calculate these separately:
(27)^(1/6) = 27^(1/6) = 3
((27/4)^(1/2)) = (√(27/4)) = (√27/√4) = (√27/2) = (√(9*3)/2) = (√9*√3/2) = (3√3/2)
Now, substitute these values back into the original expression:
(₆√27 - √27/4)^2 = (3 - 3√3/2)^2
To square this expression, we need to expand it using the binomial formula: (a - b)^2 = a^2 - 2ab + b^2
(a = 3, b = 3√3/2):
(3 - 3√3/2)^2 = (3^2) - 2(3)(3√3/2) + (3√3/2)^2
= 9 - 18√3/2 + 9/4 * 3
= 9 - 9√3 + 27/4
Now, simplify:
9 - 9√3 + 27/4 = 36/4 - 9√3 + 27/4
= (36 + 27)/4 - 9√3
= 63/4 - 9√3
Therefore, the value of (₆√27 - √27/4)^2 is 63/4 - 9√3.