Simplify: ( 7/2+5/6 )^2 - ( 7/2-5/6 )^2
7 / 2 = ( 7 / 2 ) * ( 3 / 3 ) = 21 / 6
( 7 / 2 + 5 / 6 ) ^ 2 - ( 7 / 2 - 5 / 6 ) ^ 2 =
( 21 / 6 + 5 / 6 ) ^ 2 - ( 21 / 6 - 5 / 6 ) ^ 2 =
( 26 / 6 ) ^ 2 - ( 16 / 6 ) ^ 2 =
676 / 36 - 256 / 36 =
( 676 - 256 ) / 36 =
420 / 36 =
( 12 * 35 ) / ( 12 * 3 ) = 35 / 3
Or, since it is the difference of two squares,
(7/2+5/6)^2 - (7/2-5/6)^2
= (7/2+5/6 + 7/2-5/6)(7/2+5/6 - 7/2-5/6)
= (7)(5/3)
= 35/3
To simplify the given expression, we'll follow these steps:
Step 1: Simplify the expressions within each set of parentheses.
Step 2: Evaluate the exponents.
Step 3: Subtract the second squared expression from the first squared expression.
Let's start with Step 1:
First set of parentheses: (7/2 + 5/6)
To add these fractions, we need to find a common denominator, which is 6. We can do this by multiplying the first fraction by 3/3 and the second fraction by 2/2. This gives us:
(7/2 + 5/6) = (21/6 + 5/6) = (26/6)
Second set of parentheses: (7/2 - 5/6)
To subtract these fractions, we again need to find a common denominator, which is 6. We can do this in the same way as before:
(7/2 - 5/6) = (21/6 - 5/6) = (16/6)
Now, let's move on to Step 2:
Square the expressions we obtained from both sets of parentheses:
(26/6)^2 = (26^2)/(6^2) = (676/36)
(16/6)^2 = (16^2)/(6^2) = (256/36)
Lastly, in Step 3, we subtract the latter expression from the former:
(676/36) - (256/36) = (676 - 256)/(36) = 420/36
Now, let's simplify this fraction further:
We can simplify the numerator by dividing both 420 and 36 by their greatest common divisor (GCD), which is 12:
(420/12)/(36/12) = 35/3
Therefore, the simplified form of the expression (7/2 + 5/6)^2 - (7/2 - 5/6)^2 is 35/3.