## Melodious one year ago which is smaller 5 root 10 or 4 root 9?????????

1. anonymous

You mean $$5\sqrt{10}$$ or $$4\sqrt{9}$$? Observe that $$5>4$$ and $$\sqrt{10}>\sqrt{9}$$. Which do you think is larger?

2. Melodious

4 root 9 is larger right?

3. anonymous

No. Since 4<5 and sqrt9 < sqrt10, one would expect that 4sqrt9<5sqrt10, otherwise one would violate some variant of Archimedes' principle.

4. Melodious

can u simplify the roots and help me out @nettle404

5. anonymous

I think $$\sqrt{10}$$ is about as simple as it can get. On the other hand, what squared gives you $$9$$?

6. Melodious

3

7. anonymous

Exactly, so $$\sqrt9=3$$. But $$\sqrt{10}$$ has no simple expression.

8. Melodious

but the question is 4 root 9 so it means 4 multiplied by root 9

9. Melodious

which is 3x4=12

10. anonymous

Yep.

11. anonymous

You got it.

12. Melodious

13. anonymous

Not easily simplified. Can you think of any way to square a number to obtain 10?

14. zepdrix

You can introduce another number like this if it helps you understand what is going on. $\large\rm 4\sqrt9\lt5\sqrt9$But also$\large\rm 5\sqrt9\lt5\sqrt{10}$So we see that$\large\rm 4\sqrt{9}\lt5\sqrt9\lt5\sqrt{10}$Therefore$\large\rm 4\sqrt9\lt5\sqrt{10}$

15. zepdrix

Maybe that's more complicated though :3 lol

16. anonymous

How about proof by contradiction? Suppose that $$5\sqrt{10}\leq4\sqrt{9}$$, then $$5\sqrt{10}\leq12$$ and $$\sqrt{10}\leq12/5$$. Since $$\sqrt{10}>3$$ but $$12/5<3$$, we have $$3<3$$, which is false.

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