quadratics

[bstalling]

[sqrt(9 + sqrt(80)) + sqrt(9 - sqrt(80))]^2 =

a) 1
b) 9- 4sqrt(5)
c) 18- 4sqrt(5)
d) 18
e) 20

OA - E

[truplayer256]

I bet there's an easier way to solve this but this is how I did it:

[sqrt(9+sqrt80)+sqrt(9-sqrt80)] * [sqrt(9+sqrt80)+sqrt(9-sqrt80)]
=9+sqrt(80)+2sqrt(9+sqrt80)*sqrt(9-sqrt80)+9-sqrt80
sqrt 80's cancel and you're left with:
18+2sqrt(9+sqrt80)*sqrt(9-sqrt80)
Since you're multiplying two radicals, you can multiply the inside terms and make it into one radical.
18+2sqrt(81-80)=18+2=20 E

[Mustang]

I found a slightly easier way

the original question is in the form: (a+b)^2 where a= sqrt (9+sqrt(80))

and b is : sqrt (9-sqrt(80))

(a+b)^2= a^2 + b^2 + 2*a*b

Substituting the above values we get

9+ sqrt(80)+9 - sqrt(80) + 2 *[sqrt(9+ sqrt(80)) *sqrt(9-sqrt(80))]

= 18 + 2*[sqrt{ (9+ sqrt(80)) *(9-sqrt(80))} ]

the expression within [] is of form (a+b)(a-b) = a^2 -b^2

a=9

b= sqrt(80)


solve it to get answer 20

[sureshbala]

Folks, this can be answered quite easily without getting into too much of calculation.

Look at this....

[mdavis]

How do you multiply sqrt (a+b) by sqrt (a+b)?

Also, how do you multiply sqrt (a+b) by sqrt (a-b)?

I like the above explanation but am having a difficult time multiplying square roots when there is more than one term under the sign. Thanks in advance!

[BuckeyeT]

mdavis,

To your first question -
For like termed exponents, add the terms when multiplying. For example, (2^n)(2^m)=2^(n+m)

Specifically, sqrt (a+b) can be rewritten as (a+b)^(1/2). So, sqrt (a+b) * sqrt (a+b) = [(a+b)^(1/2)] * [(a+b)^(1/2)] = (a+b)^((1/2)+(1/2)) = (a+b)^1 = a+b.

To your second question -
As truplayer256 mentioned,

Since you're multiplying two radicals, you can multiply the inside terms and make it into one radical.

So, multiply out the terms and slap them inside the radical (sqrt). sqrt (a+b)*sqrt (a-b) = sqrt (a^2-ab+ab+b^2) = sqrt (a^2+b^2).

Does that help explain it a bit better?

[sureshbala]

How do you multiply sqrt (a+b) by sqrt (a+b)?

Also, how do you multiply sqrt (a+b) by sqrt (a-b)?

I like the above explanation but am having a difficult time multiplying square roots when there is more than one term under the sign. Thanks in advance!

Dear friend,

We have the following simple rule regarding square roots.
sqrt(p) x sqrt(q) = sqrt(pq)

So sqrt(a+b) x sqrt(a-b) = sqrt[(a+b)(a-b)] = sqrt[a^2 - b^2]. I applied the same rule in my explanation