Comparing, which Is Larger? Learn this trick!

9^63 and 6^93, which one is larger? A fantastic math problem. A common method to this kind of questions.

[mathwindow]

If you have any suggestions or questions on math, comment as a reply! ❤😊

[devondevon]

3^21
This turns out easy 2^31 vs. 3^21
2^31 = ( 2^3)^10.333 = 8^10.333

3^21 = (3^2)^10.5 = 9^10.5
9 is greater than 8, and 10.5 is greater than 10:333; ie, both the base and the exponent are greater
than the other value.

[nickcellino]

Very clever solution. I love "which is larger" problems.

[generaclesdey]

Use LOG: Log of 3 (0.4771) times 21 is about 10. Log of 2 (0.3010) times 31 is about 9.33. QED!!

[JoeTaxpayer]

8^(10-1/3) vs 9^(10-1/2) base and exponent both higher in second expression.

[reinhardtristaneugen]

my fast- approach for I am suffering by asymmetrics and by a weather that is crap to my microthermalicity: I divided the powers by 10, cause you won't take away the proportions in doing so, and then you round both exponents to the full integer of the powers of 3 and 2 respectively and you'll get that of course 9 > 8 and you can infer dead sure by this resulat that these proportion will last through the whole process of calculating up to 21 or 31 respectively... ...thus 3 to the power of 21 is bigger but it is not that much bigger just a little bit...
Le p'tit Daniel

[shennalim]

Very clear. I like your use of coloured pens to show your working.

[haihe]

If you divide both sides by 2^21, we are literally comparing 2^10 and (3/2)^21 without changing the sign of the inequality. Notice that 3/2 = 1.5, we (3/2)^21 = (1.5)^(1 + 20) = (1.5) * (1.5)^20 = 1.5 * [(1.5)^2]^10 = 1.5 * [2.25]^10. So we are comparing 2^10 and 1.5 * (2.25)^10. We can see that, 1.5 * (2.25)^10, or (3/2)^21, is greater than 2^10. In other words, 3^21 > 2^31.

[KenFullman]

Without calculating the actual value 3^21 has 11 digits, whereas 2^31 only has 10 digits. This tells us that 3^21 is the larger value.
I think this is a much quicker way at reaching the answer required.

[jim]

31/21 = 1.47 which slightly less than 3/2. Approximations: 2^(3/2) = √8 = 2√2 = 2 x 1.41 = 2.82 which is less than 3.

[devondevon]

Say you have an exponent and a base say 5^6 this is the same as (5^2)^6/2 = 25^3

it is also (5^3)^6/3 = 125^3 , iti s also( 5^4)^ 6/4 - = 625^1.5 whatever you increase the
exponent by you have to compensate by dividing the base by

[rudolfsverdins]

Before I watch it: 2*8^10<3*9^10. Sort of trivial?

[dailymoonpie]

What do random numbers prove? You cannot even use it to generate a generic formula. It's just a waste of time

[zembalu]

x^y = 10^(y*lg x). Let us find the smallest y*lg x. Voilá!