Which of the following numbers is greater?

Solution 1:

Note: $7^2<2\cdot 5^2$ and $5>2^2$

$7^{55}<7\cdot 5^{54}\cdot 2^{27}<5^{55}\cdot 2^{28}<5^{69}<5^{72}$ as required

With an extra jink into factors of $3$, we can show $7^{55}<5^{67}$

Extra notes: $3^3>5^2$ and $5^5>3\cdot2^{10}$

$7^{55}<7\cdot 5^{54}\cdot 2^{27}<5^{54}\cdot 2^{30}<5^{52}\cdot 2^{30}\cdot 3^{3}<5^{67}$

Solution 2:

Observe that $7^2=49<2\cdot 5^2$. In this case, $$ 7^{55}=7\cdot 7^{54}=7\cdot(7^2)^{27}<7\cdot (2\cdot 5^2)^{27}=7\cdot 2^{27}5^{54}. $$ Observe that $2^3<10=2\cdot 5$. In this case, $$ 7\cdot 2^{27}5^{54}=7\cdot (2^3)^95^{54}<7\cdot(2\cdot 5)^95^{54}=7\cdot 2^9\cdot 5^{63} $$ Using that $2^3<10=2\cdot 5$ again, we get $$ 7\cdot 2^9\cdot 5^{63}=7\cdot (2^3)^3\cdot 5^{63}<7\cdot (2\cdot 5)^3\cdot 5^{63}=7\cdot 2^3\cdot 5^{66}. $$ Since $7\cdot 2^3=7\cdot 8=56<125=5^3$, we get $$ 7\cdot 2^3\cdot 5^{66}<5^{69}<5^{72}. $$

Solution 3:

Okay you have $A = \frac {7^{55}}{5^{72}}=(\frac 75)^{55}\times(\frac 15)^{17}$

well, since you choose to go that way:

$= (\frac {49}{25})^{27}(\frac 1{5})^7\times [\frac 7{5}]$

$= (2*\frac{49}{50})^{27}(\frac 14\times \frac45)^{17}[\frac 7{5}]$

$=2^{27}\times2^{-34}\times[(\frac{49}{50})^{27}\times (\frac 45)^{17}\times \frac 75]$

As $\frac {49}{50} < 1$ and $\frac 45 < 1$ then

$< 2^{27}\times2^{-34}\times \frac 75$

$= \frac 7{2^7*5} < 1$.

So $7^{55} < 5^{72}$. By quite a lot actually.

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Another way of doing it would be

$\log 7^{55} = 55 \log 7$ and $\log 5^{72} = 72\log 5$

And $\log 7 = \frac 12 \log 49 \approx^- \frac 12 \log \frac{100}2 = 1 - \frac {\log 2}2$. So $\log 7^{55} \approx^- 55 - 22\frac 12 \log 2$

$\log 5 = \log \frac {10}2 = 1 - \log2$. So $\log^{72} = 72 - 72\log 2$

So $7^{55} ??? 5^{72}$

if $55 - 22\frac 12 \log 2 ???^- 72 - 72\log 2$

$49\frac 12 \log 2 ???^- 17$

And $\log 2 = \frac 1{10} \log 2^{10} = \frac 1{10} \log 1024 \approx^+ \frac 3{10}$

So $49\frac 12 \log 2 \approx^+ 4.95\times 3 < 17$.

There is a bit of margin of error as $\log 2 > 3/10$ and $\log 7 < 1 - \frac {\log 2} 2$ but the margin is not significant.

Solution 4:

We have $7^4 = 49^2 < 50^2 = 4 \times 5^4 < 5^5$. Hence $$7^{55} < 7^{56} = (7^4)^{14} < (5^5)^{14} = 5^{70} < 5^{72}.$$