Expand and simplify the following; (2 + 3^0.5)^2 - (2 - 3^0.5)^2

(2 + 3^0.5 )^2 - (2 - 3^0.5 )^2

Firstly we will start by seperating the 2 brackets. This will give us 

1 :      (2+ 3^0.5 )^2

2 :      (2 - 3^0.5 )^2

We will start by expanding our brackets. To do this we use the rule;

(a + b)^2 = a^2 + 2ab + b^2

By looking at the first set of brackets we can see that a = 2 and b = 3^0.5 

This then gives us 2^2 + 2*2*3^0.5 + (3^0.5)^2

The (3^0.5)^2 will become 3. We know this as if you square a root you will have the original number. So for this the square root of 3 squared is 3. 

This then simplifies the first brackets to become 4 + 4*3^0.5 + 3 which becomes 7 + 4*3^0.5

Next we look at the second bracket, (2 - 3^0.5)^2

Again we use the rule (a + b)^2 = a^2 + 2ab + b^2 with a = 2 and b = -3^0.5

expanding the brackets then gives us 2^2 + 2*2*-3^0.5 + (-3^0.5 )^2

By using the same method as earlier we know that (-3^0.5)^2 will simplify to 3. However with this one we have a minus inside the brackets, this will become a positive as if you square a negative number it is always positive.

So from this 2^2 + 2*2*-3^0.5 + (-3^0.5 )^2 becomes 4 -4*3^0.5 + 3 which again shortens to 7 - 4*3^0.5.

Now we have simplified our two parts we can put them back into the original equation giving;

7 + 4*3^0.5 - (7 - 4*3^0.5)

We put brackets around the 2nd section as it still is minused from the first section, the brackets allows us to make sure we minus all of the 2nd section not just the first term. 

Now simplifying the two brackets by negating the 2nd from first gives us 7 + 4*3^0.5 - 7 + 4*3^0.5 which shortens to become 8*3^0.5. This is the simpliest answer as if we multiply 8 by root 3 we will get an unrational number.

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