Finding κ in (√3)^7 + (√3)^5 + (√3)^3 + 42(√3) = 3^κ

[styxrihocc]

Given that (√3)^7 + (√3)^5 + (√3)^3 + 42(√3) may be expressed as 3^κ,
find the value of κ


Now I know that (√3)^ 7 is the same as 3^3.5 so I have 3.5, 2.5 and 1.5 but i have no idea how to calculate 42 (√3). Please help :(

 

[eumyang]

Given that (√3)^7 + (√3)^5 + (√3)^3 + 42(√3) may be expressed as 3^κ,
find the value of κ


Now I know that (√3)^ 7 is the same as 3^3.5 so I have 3.5, 2.5 and 1.5 but i have no idea how to calculate 42 (√3). Please help :(

Factor out a square root of 3.
$(\sqrt{3})^7 + (\sqrt{3})^5 + (\sqrt{3})^3 + 42(\sqrt{3}) = \sqrt{3}(\text{...} + \text{...} + \text{...} + 42)$
The expression inside the parentheses will simplify nicely.

 

[styxrihocc]

I'm sorry i don't get it, how is that supposed to solve the problem? would be great if you could explain in detail

 

[eumyang]

I'm sorry i don't get it, how is that supposed to solve the problem? would be great if you could explain in detail

Without giving it away, I am factoring out the greatest common factor. If you have a polynomial like this:
x⁴ - 3x³ - 11x² + 33x

The GCF would be x, so factoring it out would give you this:
x(x³ - 3x² - 11x + 33)

In the same way, the GCF of the expression you have is the sqrt root of 3, so factor it out. What is the resulting expression inside the parentheses?

 

[styxrihocc]

√3(√3^6+√3^4+√3^2+42)
so how do i find κ?

 

[eumyang]

√3(√3^6+√3^4+√3^2+42)
so how do i find κ?

Now simplify what is in the parentheses.
What is
$(\sqrt{3})^6$?
$(\sqrt{3})^4$?
$(\sqrt{3})^2$?

 

[styxrihocc]

27+9+3+42 so 81 total...what now??

 

[eumyang]

So you have
$81\sqrt{3}$
Write 81 as a power of 3.
81 = 3^?
And you know that
$\sqrt{3} = 3^{1/2}$.
Put it all together...?

 

[styxrihocc]

damn now i feel really stupid having asked the question because that makes perfect sense. thanks a lot man