Inspired by Nihar Mahajan

Algebra Level 2

$\Large \color{#D61F06}{a}^{\color{#20A900}{c}} > \color{#3D99F6}{b}^{\color{#20A900}{c}}$

Given that $\color{#D61F06}{a},$ $\color{#3D99F6}{b},$ and $\color{#20A900}{c}$ are real numbers such that

$\color{#D61F06}{a} \color{#3D99F6}{b} > 0$ .
$\color{#20A900}{c} > 0$ .
$\color{#20A900}{c}$ is not an even integer.
$\color{#D61F06}{a} > \color{#3D99F6}{b}$ .

What can be said about the very first statement above?

It is false for all values of

a, b, c

It is true for only certain values of

a, b, c

. It is true for all values of

a, b, c

2 solutions

Pranshu Gaba
Oct 8, 2015

$ab > 0$ means that either $a$ and $b$ are both positive or $a$ and $b$ are both negative.

If $a$ and $b$ are both positive, then the statement is true.
If $a$ and $b$ are both negative:
- If $c$ is an odd integer, then the statement is true.
- If $c$ is number like $\frac{ 1} { 2 }$ , then both $a ^{ c}$ and $b ^{ c}$ will not be real numbers and we cannot compare non-real numbers, so the statement becomes false.

Hence the statement is $\boxed { \text { True for only certain values of } a, b, c.}$ $_\square$

It seems a and b has to be positive. Otherwise, last statement can't be written. (inequalities don't hold for complex numbers).

Harish Sasikumar - 5 years, 8 months ago

I totally agree with you; Inequalities don't hold for the complex numbers, that is why the last statement is false.

Consider $a = -1, b = -4, c = \frac{ 1} { 2}$ . Now the statement becomes

$i > 2i$

which is incorrect, and is false. But the statement is true for $a = -1, b = -4, c = 1$ . So it is true for only certain values of $a, b, c$ and false otherwise.

Pranshu Gaba - 5 years, 8 months ago

How did u clear kvpy.I mean how did u study and what did u focused on more

kishan chaudhary - 5 years, 8 months ago

I mainly studied from NCERT books, and also solved the past papers of KVPY. I focused more on physics and maths and less on chemistry and biology.

Pranshu Gaba - 5 years, 8 months ago

but u had finished your 11th portion in tenth only so doesnt that give you an advantage

kishan chaudhary - 5 years, 7 months ago

@Kishan Chaudhary – I had finished my 11th portion in tenth. How did you know that?

Yes, that gives me an advantage.

Pranshu Gaba - 5 years, 7 months ago

@Pranshu Gaba – i am in resonance borivali. i have seen you asking doubts to rahul sir. and i thought you were a genius studying university physics book and asking doubt which even rahul sir coudnt solve.

kishan chaudhary - 5 years, 7 months ago

@Kishan Chaudhary – if i ask my doubts on brilliant will u solve it for me in simple manner please

kishan chaudhary - 5 years, 7 months ago

@Kishan Chaudhary – I am sorry but my exams are going on right now and I am not getting much free time.

Pranshu Gaba - 5 years, 7 months ago

@Pranshu Gaba – its ok no problem

kishan chaudhary - 5 years, 7 months ago

Xiaoying Qin
Oct 9, 2015

a can be -1 or something like that. b would then be less than that, maybe -2. If you have a positive exponent like 0.1 then a would be -(1)^0.1 and b would be (-2)^0.1, and a would be -1 and b would approximately be -1.077.

Inspired by Nihar Mahajan

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