Articles of algebra precalculus

Why is $i^i$ real?

Possible Duplicate: How to raise a complex number to the power of another complex number? My calculator (as well as WolframAlpha) gives me the approximation: $$0.2078795763507619085469…$$ But I don’t understand how exponentiating two purely imaginary constructs yields a real (albeit irrational) number. When I do $i^{i+1}$ it gives me an imaginary number as well as […]

Finding the zeroes of the function.

I would like to find the zeroes of the following function given that $3-i$ is a zero of $f$: $f(x) = 2x^4-7x^3-13x^2+68x-30$ Please explain to me how to do this problem. Thanks!

How to make four 7 s equal to 4 and to 10?

How to make four 7 s equal to 4 and to 10? f(7, 7, 7, 7) = 1: 7/7 * 7/7 = 1 f(7, 7, 7, 7) = 2: 7/7 + 7/7 = 2 f(7, 7, 7, 7) = 3: (7+7+7)/7 = 3 f(7, 7, 7, 7) = 4: ? . . f(7, 7, 7, […]

Advanced Algebraic Equation – Solve for P

Please solve the following equation and leave your answer in terms of P. This problem has been bugging me for months now, and I have not been able to reduce it past this form: $$ (P/(L-P))^K * ((P-K)/P)^L = Ae^({KT}(LK)(L-K)) $$ where $A = e^c$ and c is some constant. I have tried to solve […]

If $f$ is a strictly increasing function with $f(f(x))=x^2+2$, then $f(3)=?$

Bdmo 2014 regionals(a tweaked version of question): If $f$ is a strictly increasing function over the reals with $f(f(x))=x^2+2$, then $f(3)=?$ Obviously,$f(3)=f(1)^2+2$ but I can’t see where we are going to use the ‘strictly increasing’ fact.I don’t think there is a way to reverse-engineer such a function without heavy machinery.I have plugged in loads of […]

Integral with quadratic square root inside trigonometric functions

Is there anyway to solve $\displaystyle \int t \frac{\sin \left(\frac{t}{2} \sqrt{ a \left(t+ \frac{b}{2a}\right)^2-\frac{b^2-4ac}{4a}}\right) }{ \sqrt{ a \left(t+ \frac{b}{2a}\right)^2-\frac{b^2-4ac}{4a}}} \operatorname{d}t \tag8$ either by analytical method or from geometrical method with out using numerical methods? means looking for a closed form with out infinite series expansion in the result NB: Main issue is the lack of […]

Combination of quadratic and cubic series

I’m an eight-grader and I need help to answer this math problem (homework). Problem: Calculate $$\frac{1^2+2^2+3^2+4^2+…+1000^2}{1^3+2^3+3^3+4^3+…+1000^3}$$ Attempt: I know how to calculate the quadratic sum using formula from here: Combination of quadratic and arithmetic series but how to calculate the cubic sum? How to calculate the series without using calculator? Is there any intuitive way […]

Showing $(a+b+c)(x+y+z)=ax+by+cz$ given other facts

$$x^2-yz/a=y^2-zx/b=z^2-xy/c$$ None of these fractions are equal to 0.We need to show that, $(a+b+c)(x+y+z)=ax+by+cz$ This question comes from a chapter that wholly deals with factoring homogeneous cyclic polynomials.I multiplied the three sides of the first equality by $abc$ but that yields an unfactorizable polynomial.I haven’t had much luck in manipulating the first equality.So I tried […]

How to show that $a,\ b\in {\mathbb Q},\ a^2+b^2=1\Rightarrow a=\frac{s^2-t^2}{s^2+t^2},\ b= \frac{2st}{s^2+t^2} $

I want show the following $$a,\ b\in {\mathbb Q},\ a^2+b^2=1\Rightarrow a=\frac{s^2-t^2}{s^2+t^2},\ b= \frac{2st}{s^2+t^2},\ s,\ t\in{\mathbb Q} $$ How can we prove this ? [Add] Someone implies that we must use pythagorean triple : Let $$ a=\frac{n}{m},\ b= \frac{s}{k},\ (m,n)=(s,k)=1$$ Then $$ k^2n^2+s^2m^2=m^2k^2 \Rightarrow n^2|(k^2-s^2),\ k^2|m^2 $$ so that we have $$n^2+ s^2=k^2,\ (n,s)=1,\ k=m$$ We […]

If $a,b,c$ are positive integers, with $a^2+b^2-ab=c^2$ prove that $(a-b)(b-c)\le0$.

I have an inequality problem which is as follow: If $a,b,c$ are positive integers, with $a^2+b^2-ab=c^2$ prove that $(a-b)(b-c)\le0$. I am not so good in inequalities. So, please give me some hints so that I can proceed. Thanks.