Surds leading to yet more confusion.....?

Question

Can someone explain how the following works please:

1) = 2 + 3 + 3 root 5 - root 5
= 5 + 2 root 5 (I'm not sure where 5 + 2 comes from, I
would've thought it'd be 5 + 3)

2) = 4 + 2 root 3 + 2 root 3 +3
= 7 + 4 root 3 (I know 4 + 3 = 7 and 2 + 2 = 4 but why is it
added in this way?)

3) = root 2 x root 2 + root 2 x 2 root 3 x root 2 + root 3 x 2 root 3
= 2 + 2 root 6 + root 6 + 2 x 3
= 8 + 3 root 6 (A lot of roots sorry, but very confused!)

4) = 2 x 2 - 2 x root 3 + root 3 x 2 - root 3 x root 3
= 4 - 2 root 3 + 2 root 3 - 3
= 1

Any help please would be very greatly appreciated!! :o)

Answer 1

I will use your first problem for explanation, the rest should be treated similarly.

2 + 3 + 3 root 5 - root 5 = 5 + 2 root 5 Let's cut this up with brackets (tricky on this computer as it isn't mine)

(2 + 3) + (3 root 5) - root 5 = (5) + (2 root 5)

now add some multiplication signs...
(2 + 3) + (3 * root 5) - root 5 = (5) + (2 * root 5)

now you can see where the different parts come from... it is not 2+3+3 but 2+3 + (3 * 2.2360)

You have to learn that the multiplication sign is not always shown in "higher" mathematics. 3Y = 3*Y this is because the multiplication symbol (x) can get confused with the "Constant" X and until computers became common place, nobody used the asterisk in its place.

This applies in (2) as well it's 7+ (4 x √3) so (3+4) + (2 x √3) + (2 x √3)

(3) √2 x √2 = 2; √3 x √3 = 3 ... 2 x 3 = 3 x 2 ...

Any further problems, speak to your maths teacher for further clarification.

Answer 2

1) = 2 + 3 + 3 root 5 - root 5
= (2+3) + (3 root 5 - root 5 )
= 5 + (2 root 5)
You do not have to show the brackets because based on BoDMAS, you multiply first when you calculate this, i.e. 2 x root5, so writing 5 + 2root5 is correct.

2) 4 + 2 root 3 + 2 root 3 +3
= (4+3) + (2root3 + 2 root 3)
= 7 + 4 root 3
As in algebra, you cannot add x terms to y terms, i.e. x + 2y is NOT "add-ible", so similarly, if you want your answer to remain in surd form, you cannot add it to a non-surd or to surds with different numbers, i.e root 2 cannot be added to root 3 and remain a surd.

3) root 2 x root 2 + root 2 x 2 root 3 x root 2 + root 3 x 2 root 3
= (root2 )^2 + (root2 x root2 x 2 x root3) + (root3 x 2 x root3)
= 2 + 4 root3 + 6
= 8 + 4 root3

Your answer is incorrect.
You have to realise that root 2 x root 2 = (root 2)^2 = 2.
Similarly, root 3 x root 3 = 3.

4) 2 x 2 - 2 x root 3 + root 3 x 2 - root 3 x root 3
= 4 - 2 root 3 + 2 root 3 - 3
= 4 - 3
= 1

Same explanation as in 3).

Answer 3

Only like terms are added. I am doing your one question
1) = 2 + 3 + 3 root 5 - root 5
= (2+3) + (3 root 5 -root 5)
=(2+3) + root 5(3-1) (Taking root 5 common)
= 5 + 2 times root 5

Answer 4

You have to understand the rules of indices.

a square root of a number is equivalent to the number raised to the power of 1/2

a cube root is 1/3 etc

Check out the link below for rules of indices.

Answer 5

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Answer 6

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Answer 7

A lot of roots there indeed. Here's one more. ;)