Factor Expression: A2+B2 from 3[√3+√5+√7]2

[cupcakes]

Homework Statement

Express the following as A²+B²:

3[√3+√5+√7]²

Homework Equations

The Attempt at a Solution

I expanded it to 45 + 6√15 + 6√21 + 6√35
Should I collect like terms (the multiples of 6)? I don't know how to proceed from here. Thanks in advance for any help.

 

[tal444]

I'm not sure, but I don't think this expression can be converted to the form A²+B².

 

[cupcakes]

Maybe it's because I forgot to mention that A & B may contain roots. I think it's possible because it is an assigned question. But I'm stuck...

 

[e^(i Pi)+1=0]

There are no real numbers that factor to the sum of two squares. Your answer will be imaginary.

 

[chubbyorphan]

3[√3+√5+√7]^2

3[√(1.5)+√(2.5)+√(3.5)]^2 + 3[√(1.5)+√(2.5)+√(3.5)]^2

up in the middle of the night doing this.. so I may have broken a million rules getting to this point
You Might want to double check but they seem equivalent

 

[Mark44]

How about this?
3[√3+√5+√7]² = 2[√3+√5+√7]² + [√3+√5+√7]²

The original expression is now written as a sum of two terms. Can you finish the problem by showing that each of these terms is the square of something? I.e., can you identify A and B with the above being equal to A² + B²?

 

[chubbyorphan]

It only works with odd numbers

(√1.5 + √3.5)² + (√1.5 + √3.5)² = (√3 + √7)²

The left side is 2 * 9.58 = 19.16. The right side is 19.16(√44.5 + √6.5)² + (√44.5 + √6.5)² = (√89 + √13)²

The left side is 2 * 85.014 = 170.029. The right side is 170.029

yesh?

 

[Mark44]

It only works with odd numbers

(√1.5 + √3.5)² + (√1.5 + √3.5)² = (√3 + √7)²

The left side is 2 * 9.58 = 19.16. The right side is 19.16


(√44.5 + √6.5)² + (√44.5 + √6.5)² = (√89 + √13)²

The left side is 2 * 85.014 = 170.029. The right side is 170.029

yesh?

Yes, this works. My example (now deleted) was flawed in that I forgot to square the value on the right side. Apologies for the misdirection...

Here's what's going on.
(√1.5 + √3.5)² + (√1.5 + √3.5)² = 2(√1.5 + √3.5)²
= (√2 *√1.5 + √2 *√3.5)²
= (√3 + √7)²

 

[chubbyorphan]

How about this?
3[√3+√5+√7]² = 2[√3+√5+√7]² + [√3+√5+√7]²

The original expression is now written as a sum of two terms. Can you finish the problem by showing that each of these terms is the square of something? I.e., can you identify A and B with the above being equal to A² + B²?

Hey no worries, Mark44, beside I found my version of the solution completely out of luck just bored and messing around with my calculator :P
You're version of the sum of two terms is just as valid isn't it?

 

[cupcakes]

How about this?
3[√3+√5+√7]² = 2[√3+√5+√7]² + [√3+√5+√7]²

How did I not notice that?! That works perfectly Mark. Thanks to everyone for their help.