1. ## Which radical expression is larger?

$\displaystyle \sqrt[22]{10} \ \ \ or \ \ \ \sqrt[30]{30}$

The type is relatively small. That is the equivalent of 10^(1/22) versus 30^(1/30).

You may use a pencil and paper. You may not use a calculator, computer, or logarithmic tables.

2. ## Re: Which radical expression is larger?

Hint: try raising both expressions to the 30th power and see which is larger.

3. ## Re: Which radical expression is larger?

Originally Posted by ChipB
Hint: try raising both expressions to the 30th power and see which is larger.
If you do that, then you will have 10^(15/11) versus 30. What would be your next
step? I have it as a challenge problem, which means I'm challenging others.

My strategy involves getting rid of fractional exponents for both sides at the outset.

4. ## Re: Which radical expression is larger?

$10^{1/22} < 30^{1/30}$

Spoiler:
$10^{15/11} \ne 3 \cdot 10$

$10^{15} \ne 3^{11} \cdot 10^{11}$

$10^4 \ne 3^3 \cdot 9^4$

$\dfrac{10^4}{9^4} \ne 27$

$\dfrac{10000}{(80+1)^2} \ne 27$

$\dfrac{10000}{6400+160+1} < 2 < 27$

5. ## Re: Which radical expression is larger?

Originally Posted by greg1313
$\displaystyle \sqrt[22]{10} \ \ \ or \ \ \ \sqrt[30]{30}$

The type is relatively small. That is the equivalent of 10^(1/22) versus 30^(1/30).

You may use a pencil and paper. You may not use a calculator, computer, or logarithmic tables.
\begin{align} \sqrt[22]{10} &\leftrightarrow \sqrt[30]{30} \\ \frac1{22}\log{10} &\leftrightarrow \frac1{30}\log{30} \\ 30\log{10} &\leftrightarrow 22(\log{10} + \log{3}) \\ 8\log{10} &\leftrightarrow 22\log{3} \\ \frac{\log{10}}{\log{3}} &\leftrightarrow \frac{22}{8} \\ 2 = \frac{2\log{3}}{\log{3}} = \frac{\log{9}}{\log{3}} \approx \frac{\log{10}}{\log{3}} &\lt \frac{22}8 \approx 3 \end{align}

6. ## Re: Which radical expression is larger?

$\displaystyle \frac{\sqrt[30]{30}}{\sqrt[22]{10}} = \frac{\sqrt[30]{10}\sqrt[30]{3}}{\sqrt[22]{10}} = \frac{\sqrt[150]{243}}{\sqrt[165]{100}} > 1 \longrightarrow \sqrt[30]{30} > \sqrt[22]{10}$

7. ## Re: Which radical expression is larger?

Originally Posted by Zexuo
$\displaystyle \frac{\sqrt[30]{10}\sqrt[30]{3}}{\sqrt[22]{10}} = \frac{\sqrt[150]{243}}{\sqrt[165]{100}}$
It looks like you're missing a step or two between these two.

Originally Posted by Zexuo
$\displaystyle \frac{\sqrt[150]{243}}{\sqrt[165]{100}} > 1$
No, I don't see how you have justified this part of it.

8. ## Re: Which radical expression is larger?

Originally Posted by Archie
\begin{align} \sqrt[22]{10} &\leftrightarrow \sqrt[30]{30} \\ \frac1{22}\log{10} &\leftrightarrow \frac1{30}\log{30} \\ 30\log{10} &\leftrightarrow 22(\log{10} + \log{3}) \\ 8\log{10} &\leftrightarrow 22\log{3} \\ \frac{\log{10}}{\log{3}} &\leftrightarrow \frac{22}{8} \\ 2 = \frac{2\log{3}}{\log{3}} = \frac{\log{9}}{\log{3}} \approx \frac{\log{10}}{\log{3}} &\lt \frac{22}8 \approx 3 \end{align}
Archie, (in effect) you supported that 2 < $\displaystyle \ \dfrac{log(10)}{log(3)}, \ \$ and we know that 2 = 16/8 < 22/8.

But I do not see support as to where you show the relative size of $\displaystyle \ \dfrac{log(10)}{log(3)} \$ versus 22/8.

*

10. ## Re: Which radical expression is larger?

$\displaystyle \frac{\sqrt[30]{10}}{\sqrt[22]{10}} = 10^{\frac{1}{30}-\frac{1}{22}} = 10^{-\frac{2}{165}} = 100^{-\frac{1}{165}}$

$\displaystyle \sqrt[30]{3} = \left(243^{\frac{1}{5}}\right)^{\frac{1}{30}} = 243^{\frac{1}{150}}$

The first inequality justified by the shallower root of a larger number in the numerator compared with the denominator (with both radicands greater than 1).

11. ## Re: Which radical expression is larger?

Originally Posted by greg1313
Archie, (in effect) you supported that 2 < $\displaystyle \ \dfrac{log(10)}{log(3)}, \ \$ and we know that 2 = 16/8 < 22/8.

But I do not see support as to where you show the relative size of $\displaystyle \ \dfrac{log(10)}{log(3)} \$ versus 22/8.
Well $\frac{\log{27}}{\log{3}}=3$ and $10$ is clearly much closer to $9$ than it is to $27$, so it is reasonable to assume that we have a number close to $2$. $\frac{22}{8}=2.75$ which is closer to $3$ than it is to $2$. With a bit of thought I could probably find something a little more rigorous, but it didn't seem necessary.

E.g: The geometric mean of $9$ and $27$ is $\sqrt{243}$ which is a little larger than $15$. Since $10<15$ we know that $\frac{\log{10}}{\log{3}} < \frac{\log{15}}{\log{3}} < 2.5$

12. ## Re: Which radical expression is larger?

Or, perhaps better:
$$\frac{\log{10}}{\log{3}} < \frac{\log{\sqrt{243}}}{\log{3}} = \frac{\log{243}}{2\log{3}} = \frac{\log{3^5}}{2\log{3}} = \frac{5\log{3}}{2\log{2}} = \frac52 < \frac{22}{8}$$

13. ## Re: Which radical expression is larger?

Archie, your fifth denominator has a typo. It should be 2log3. You may have noticed
that I gave that post a "Thanks."

14. ## Re: Which radical expression is larger?

$\displaystyle 10^{1/22} \ \ vs. \ \ 30^{1/30}$

$\displaystyle (10^{1/22})^{(2*11*15)} \ \ vs. \ \ (30^{1/30})^{(2*11*15)}$

$\displaystyle 10^{15} \ \ vs. \ \ 30^{11}$

$\displaystyle \dfrac{10^{15}}{10^{11}} \ \ vs. \ \ \dfrac{30^{11}}{10^{11}}$

$\displaystyle 10^4 \ \ vs. \ \ \bigg(\dfrac{30}{10}\bigg)^{11}$

$\displaystyle 10^4 \ \ vs. \ \ 3^{11}$

$\displaystyle \dfrac{10^4}{3^{12}} \ \ vs. \ \ \dfrac{3^{11}}{3^{12}}$

$\displaystyle \dfrac{10^4}{(3^3)^4} \ \ vs. \ \ \dfrac{1}{3}$

$\displaystyle \bigg(\dfrac{10}{27}\bigg)^4 \ \ vs. \ \ \dfrac{1}{3}$

$\displaystyle \bigg(\dfrac{10}{27}\bigg)^4 \ < \bigg(\dfrac{10}{20}\bigg)^4 \ = \ \bigg(\dfrac{1}{2}\bigg)^4 \ = \ \dfrac{1}{16} \ < \ \dfrac{1}{3}$

Therefore, $\displaystyle \ \ 30^{1/30} \ > \ 10^{1/22}$