Math Problem! Math Geeks come!?
ok, so i have a question about this problem so if you could answer it and tell me how you got it that would be great! so here it is
"Compare Powers"
Which is greater:
(a) 2 1 or 1 2 ?
(b) 3 2 or 2 3 ?
(c) 4 3 or 3 4 ?
(d) 7 6 or 6 7 ?
Predict the greater of any two positive integers that follow this pattern.
(BTW the second number of the pair like 2 1 ,1 is supposed to be an exponent)
i got the a,b,c and d but i don't get the part after it. so, if you can help me thanks!
Comments
(The usual way of indicating exponents, without special characters, is a^b.)
(a) 2^1 or 1^2 ?
For bases (the numbers that get raised to various powers) that are greater than 1, a larger exponent tends to show where the larger number is. If the bases and exponents are not close to each other, this will not always be true: 3^7 is greater than 7^3, for instance. If they are close, however, bet on the larger exponent.
1^x is a special case: 1 to any power is just 1, so 2^1 > 1^2. (2 > 1.)
(b) 3^2 or 2^3 ?
3^2 = 9; 2^3 = 8, so 3^2 > 2^3.
With small bases, y=x^2 can be larger than y=x^3.
(c) 4^3 or 3^4 ?
4^3 = 64; 3^4 = 81, so 4^3 < 3^4.
y=x^4 increases notably more rapidly than y=x^3 when x>1.
(d) 7^6 or 6^7 ?
7^6 = 117,649; 6^7 = 279,936, so 7^6 < 6^7.
y=x^6 increases rather rapidly, but y=x^7 increases even more rapidly.
When the bases (call them a and b) are larger than 2 and 3, you can count on a^b being less than b^a.
a) 2^1 > 1^2
b) 3 ^2 > 2^3
c) 4^3 > 3^4
d) 7^6 >6^7
The first of the pair will be greater
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