Solving Exponent Problems: Can't Tell Which is Bigger?

[theumann]

This problem was found in my gre test practice guide and i am having trouble understanding why the answer i chose was wrong.

x^2 =16
y^3= 64

Which is bigger x or y?
I chose that they were both equal to 4 however the answer stated that the relationship cannot be determined from the information given.

Help me understand this.

 

[DeadWolfe]

x^2 = 16 could allow x = -4

 

[mathman]

Further complication is that y^3=64 has two complex roots as well as the real root 4.

 

[HallsofIvy]

Silly problem anyway!

 

[Robokapp]

Further complication is that y^3=64 has two complex roots as well as the real root 4.

Are you sure? y is just y...so when you expand it (stating the obvious here, don't laugh) you get

y*y*y=64

Can 3 complex identical numbers give a 64? I thought they always come in groups of 2 to cancel each other out...i mean an odd power to the i in each factor would result in...and odd power in the final result...and 64 is having in best case scenario a i to a power that is multiple of 4.

(i^4 * 4) ^3 =1 * 64 for example.

So I don't think y^3=64 can get you a complex i. Can it? now I'm curious. My math says no...

 

[d_leet]

So I don't think y^3=64 can get you a complex i. Can it? now I'm curious. My math says no...

First a complex number is not just some real multiple of i, ex. a*i for real a. A complex number is a number of the form
a+bi
where both a and b are real and i² = -1.

Secondly if we have y³=64 then we have
y³ - 64 = 0
And the fundamental theorem of algebra says that every polynomial equation of degree n has n complex zeroes counting multiplicites, so since we have a polynomial equation of degree 3 we are guarenteed by the fundamental theorem of algebra that this equation had 3 solutions, however in this case only one of them is real, i.e. 4, while the other two are complex.

If you have a polynomial equation with only real coefficients then and only then will all complex roots come in conjugate pairs, a+bi and a-bi.

 

[Data]

write y = re^{i\theta} with r>0, \theta \in [0, 2\pi), then your equation is

r^3e^{3i\theta} = 64e^{0i}.

Since representation of complex numbers in the way I just mentioned for y is unique, you need 3\theta to be a multiple of 2\pi (and so 0 modulo 2\pi) and you also need 0<r \in \mathbb{R} and r^3 = 64, ie. r=4. The condition on \theta gives 3 possibilities,

\theta = 0, \theta= \frac{2\pi}{3}, \theta = \frac{4\pi}{3},

so your 3 complex solutions are 4, and y = 4(\cos{4\pi /3} + i\sin{4\pi /3}) = -2 + 2\sqrt{3}i and its complex conjugate (keep in mind that if one complex number is the root of a polynomial with real coefficients then its conjugate must also be).

 

[Robokapp]

D-Leet I understand what you mean.

y^3-64=0 turns into
(y-4)(y^2+4y+16)=0

y^2+4y+4=0 => (-4+/- sq(1-64))/2 => -2+31.5i and -2-31.5i

however, -2+31.5i, -2-31.5i and 4 are not the same answer...Shouldn't they be?

Edit: What I really am asking is...can you tell if 4 is greater or less than -2+31.5i? How do you compare real with complex numbers?

 

[HallsofIvy]

What I really am asking is...can you tell if 4 is greater or less than -2+7.5i? How do you compare real with complex numbers?

You can't. There is no way to define "< " to make the complex numbers an ordered field.

Ordered field: A field (usual properties for addition, multiplication) with a transitive relation "<" (transitive: if a< b and b< c, then a< c) satisfying
a) If a< b then a+ c< b+ c
b) If a< b and 0< c then ac< bc
c) For any a, b, one and only one of these must hold
i) a< b
ii) b< a
iii) a= b

Suppose we have some definition of "<" on the complex numbers
Obviously 0 is not equal to i (0 is the additive identity and i is not).

Is 0< i? If so then by (b), 0*i< i*i or 0< -1. That's possible since this doesn't have to be our usual idea of "<" on the real numbers. But then, again by (b), 0*i< -1*i so 0< -i also. But then by (a), 0+ i< -i+ i or i< 0. We can't have both 0< i and i< 0 by (c).

The only possibility left is i< 0. In that case, by (a) i+ (-i)> 0+ (-i) so 0< -i. By (b) then, 0*(-i)< (-i)(-i) or 0< -1 again. By (b) again, 0*(-i)< (-1)(-i) so 0< i. That also is impossible by (c).

however, -2+7.5i, -2-7.5i and 4 are not the same answer...Shouldn't they be?

No, of course not. Just as a quadratic equation may have two different solutions, a cubic equation may have 3 different solutions. In fact, counting "multiplicity" any n^th degree equation has exactly n solutions over the complex numbers. More correctly, any n^th degree polynomial, with complex coefficients, can be factored into n linear factors, some of which may be the same.

 

[mathman]

y^3-64=0 turns into
(y-4)(y^2+4y+4)=0

Error: should be (y-4)(y^2+4y+16)

 

[Robokapp]

Error: should be (y-4)(y^2+4y+16)

You are correct. I'll edit my post immediatelly. Sorry about that.

Edit: I worked it out, I understand it now. Thank you for the help. I was thinking about imaginary numbers, I wasn't thinknig about complex numbers...and ofcourse, the bi part won't get me where I want if I don't have an a. Thank you.