Without using a calculator can you say, which number is greater, $65^{1662}$ or $33^{1995}$?

Question

Without using a calculator can you say, which number is greater, $65^{1662}$ or $33^{1995}$?

first i thought we can say which is bigger by differences.
Like Here difference of base : 65–33=32.
Here difference of exponent : 1995–1662=333.
Difference of exponents are bigger than difference of base. so I think I should consider the biggest number by bigger difference. So i considered $33^{1995}$ as bigger number and I was right.
$33^{1995}$ is bigger then $65^{1662}$ .
But then i took another example $21^{6}$ and $81^{4}$.
Here difference of base : 81–21=60.
Here difference of exponent : 6–4=2.
Then as before here also i considered bigger number by bigger difference.
So here base has the bigger difference than exponent.
So according to the bigger difference of base i considered bigger number by bigger difference and that was 81^3.
But I was wrong at this point.
$21^{6}$ is bigger than $81^{3}$

Then What is the correct way of finding the bigger number ?

Answer 1

You can use the rules of exponentiation. When you have $21^6$ and $81^3$, these can be rewritten as $(21^2)^3$ and $81^3$, so the question becomes: which is bigger out of $21^2$ and $81$? Taking a root of both, and noting $21 > 9$ then yields your answer.

The important thing is to try to get the two numbers on the same exponents, possibly by estimating them from the correct side. As a rule of thumb, you might say that the exponent is almost always the most important element.

Answer 2

There are two things to note here:

1. $65$ is just less than $2 \times 33$
2. $\frac{1995}{1662}$ is just greater than $\frac{6}{5}$

This suggests the following approach:

$65^{1662} < 66^{1662} = 2^{1662} \times 33^{1662} = 32^{\frac{1662}{5}} \times 33^{1662}$

I'll let you take it from there.

Answer 3

If we divide both numbers by $33^{1662}$, we see that it's enough to find the bigger number between $$ \frac{65^{1662}}{33^{1662}} = \left(\frac{65}{33}\right)^{1662} \quad \text{and} \quad \frac{33^{1995}}{33^{1662}} = 33^{333}. $$ It suffices to note that $$ \left(\frac{65}{33}\right)^{1662} < 2^{1662} = 32^{1662/5} = 32^{332.4} < 33^{333} $$

Answer 4

Taking the log of both sides

$1662 \log 65 <?> 1995 \log 33$

$\log 65 < \log 66 = \log 33 + \log 2$

$1662 \log 65 < 1662 \log 33 + 1662 \log 2$

We can choose what base to use for our logs, lets choose log base 2

$5 < \log_2 33$

$1995 = 1662 + 333$

$1662 \log_2 33 + 1662 < 1662 \log_2 33 + 1665 < 1995 \log_2 33$