Articles of algebra precalculus

Find all reals $a, b$ for which $a^b$ is also real

The title is pretty much clear, but here is a more precise formulation: Find all pairs $(a,b)\in\mathbb{R^2}$ for which $a^b$ is also real. I used a CAS to solve the problem and it says that the solution is $$(a=0\land b>0)\lor \left(c_1\in \mathbb{Z}\land a<0\land b=c_1\right)\lor a>0$$ But i think the correct answer is $$(a=0\land b>0)\lor \left(c_1\in […]

How to go upon proving $\frac{x+y}2 \ge \sqrt{xy}$?

This question already has an answer here: Proving the AM-GM inequality for 2 numbers $\sqrt{xy}\le\frac{x+y}2$ 5 answers How to prove that $\frac{a+b}{2} \geq \sqrt{ab}$ for $a,b>0$? [duplicate] 3 answers

Help with understanding a proof that $f$ is bounded on $$ (Spivak)

I need help on the proof of Theorem 7-2 in Spivak: If $f$ is continous on $[a,b]$, then $f$ is bounded above on $[a,b]$. So, the proof starts with this: Let $$A= \{x:a\le x \le b \text{ and } f \text{ is bounded above on } [a,x]\}$$ The author then went on to prove that […]

Find the co-ordinates of the point on the join of two points which is nearest to the intersection of two planes

Find the co-ordinates of the point on the join of $(-3, 7, -13)$ and $(-6, 1, -10)$ which is nearest to the intersection of the planes $3x-y- 3z + 32 =0$ and $3x+2y-15z= 8$. Please give me an outline to solve the problem. Thanks.

How many different proofs are there that $a^n-b^n =(a-b)\sum_{i=0}^{n-1} a^i b^{n-1-i} $?

How many different proofs are there that $a^n-b^n =(a-b)\sum_{i=0}^{n-1} a^i b^{n-1-i} $ for positive integer $n$ and real $a, b$? You can use any techniques you want. My proof just uses algebra, summation, and induction, but if you want to use invariant sheaves over covalent topologies, that is fine. I decided that I would try […]

Why does $\sqrt{1/2} = 1/\sqrt{2}$ but $\sqrt{2/3} \neq 2/\sqrt{3}$?

I can see that $\sqrt{1/2} = 1/\sqrt{2}$ My calculator also confirms I can change the denominator and the equality still holds. But $\sqrt{2/3} \neq 2/\sqrt{3}$ Can someone explain why? I need to get the concept here.

Stuck on rearranging of this equation

I need to get from $[(1-p)f+p(1-f)](1+v)-[(1-p)(1-f)+pf] = x$ to $(2+v)(f+p-2pf)-1 = x$ but I’m stuck. I’d appreciate any tips on what I should I do after the following. $(f+p-2pf)(1+v) + (f + p – 2pf) – 1 = x$ Thanks in advance.

Quadratic equation which has rational roots

If the following quadratic equation $$qx^2+(p+q)x+bp=0$$ always has rational roots for any non-zero integers $p$ and $q$ what will be the value of $b$? My book’s solution says the value of $b$ will be $0$ or $1$. If we consider the discriminant of the equation, $$D=(p+q)^2-4bqp = p^2+2q(1-2b)p+q^2$$ then $D$ should be a perfect square […]

Absolute value of a real number

My question is: Solve: $|x-4|< a$, where $a$ belongs to the real numbers. Solve this by considering various cases depending upon whether $a$ is negative, positive or zero. What I have tried so far: If $a>0$ then: $x < a+4$ and $x>4-a$, if $a=0$ then there is no solution. My doubt is: Should I consider […]

Prove that $\frac{bc}{b+c}+\frac{ac}{a+c}+\frac{ab}{a+b} \leq \frac{a+b+c}{2}$

If $a,b,c \in R$, then prove that: $$\frac{bc}{b+c}+\frac{ac}{a+c}+\frac{ab}{a+b} \leq \frac{a+b+c}{2}$$ I can’t see any known inequality working here like $H.M.-A.M.$. Could this be solved using basic inequalities?