Exponents
Re: Exponenents
$1024=2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2
=2^{10}
=(2^{2})^{5}=4^{5}
=(2^{5})^{2}=32^{2}$
$\therefore (a,b)=(2,10),(4,5),(32,2)$
$\therefore$ There are $3 (a,b)s$ such that $a^{b}=1024$.
=2^{10}
=(2^{2})^{5}=4^{5}
=(2^{5})^{2}=32^{2}$
$\therefore (a,b)=(2,10),(4,5),(32,2)$
$\therefore$ There are $3 (a,b)s$ such that $a^{b}=1024$.
"Questions we can't answer are far better than answers we can't question"
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Re: Exponents
I think answer 4.
(a,b)=(2,10),(4,5),(32,2),(1024,1).
(a,b)=(2,10),(4,5),(32,2),(1024,1).
Re: Exponents
It's essential to clarify all the constraints of a problem precisely. Why should we not consider negative and fractional values for a and b for this problem? For example, what's wrong with
$(a, b)=(-2, 10), (-32, 2), (1048576, \frac{1}{2})$?
$(a, b)=(-2, 10), (-32, 2), (1048576, \frac{1}{2})$?