Neat Problem at MHF

[JeffM]

Over at Math Help Forum, there is a neat little problem

Without using a calculator, computer, book of math tables, slide rule, or brute force, determine which is smaller [imath]15^{11}[/imath] or [imath]7^{17}[/imath].

 

[blamocur]

Spoiler

[imath]5 \times 225^5 < 49 \times 343^5[/imath]

 

[greg1313]

Spoiler

[imath]5 \times 225^5 < 49 \times 343^5[/imath]

You're going to have to show more detailed steps
in your train of equalities and inequalities that connect to each other and justify your conclusion. This was based on your 7:28 a.m.
post as I read it from my location.

 

[BigBeachBanana]

How about [imath]65^{40}[/imath] and [imath]255^{30}[/imath]?

Spoiler: Solution

Since [math]65>64 = 4^3\\ 255<256= 4^4[/math] then [math]65^{40}>(4^3)^{40}=4^{120}\\ 255^{30}<(4^4)^{30}=4^{120}[/math]Therefore, [math]255^{30}<65^{40}[/math]

 

[blamocur]

You're going to have to show more detailed steps
in your train of equalities and inequalities that connect to each other and justify your conclusion. This was based on your 7:28 a.m.
post as I read it from my location.

Spoiler

[math]15^{11} = 15 \times \left(15^2\right)^5 = 15 \times 225^5[/math][math]7^{17} = 49 \times \left(7^3\right)^5 = 49 \times 343^5[/math][math]15 < 49[/math][math]15^2 = 225 < 343 = 7^3[/math][math]15^{11} = 15\times 225^5 < 49\times 343^5 = 7^{17}[/math]

 

[Deleted member 4993]

Did you check Greg's fishing license??!!

You gave away free cooked fish ........

 

[blamocur]

Did you check Greg's fishing license??!!

You gave away free cooked fish ........

I guess I was spoiling him

 

[greg1313]

Did you check Greg's fishing license??!!

You gave away free cooked fish ........

Subhotosh Khan, I was phishing for more information, and I found the rigor.

 

[greg1313]

How about [imath]65^{40}[/imath] and [imath]255^{30}[/imath]?

Spoiler: Solution

Since [math]65>64 = 4^3\\ 255<256= 4^4[/math] then [math]65^{40}>(4^3)^{40}=4^{120}\\ 255^{30}<(4^4)^{30}=4^{120}[/math]Therefore, [math]255^{30}<65^{40}[/math]

In your hypothetical example, I would shave the numbers down on each side
by doing this first, and then I would follow these other steps to the end:

Spoiler

\(\displaystyle 65^{40} \ \ vs. \ \ 255^{30} \)

\(\displaystyle (65^{40})^{0.1} \ \ vs. \ \ (255^{30})^{0.1} \)

\(\displaystyle 65^4 \ \ vs. \ \ 255^3 \)

\(\displaystyle \dfrac{65^4}{65^3} \ \ vs. \ \ \dfrac{255^3}{65^3}\)

\(\displaystyle 65 \ \ vs. \ \ \bigg(\dfrac{5\cdot 51}{5 \cdot 13} \bigg)^3 \)

\(\displaystyle 65 \ \ vs. \ \ \bigg(\dfrac{51}{13}\bigg)^3 \)

\(\displaystyle (\tfrac{51}{13})^3 < (\tfrac{52}{13})^3 = 4^3 = 64 \)

\(\displaystyle 65 > 64, \ so \ \ 65 > (\tfrac{51}{13})^3. \)

\(\displaystyle Therefore, \ \ 65^{40} > 255^{30}. \)