# Which side has a larger value?

#### checkittwice

##### Member
Without using the square root button on a calculator,
determine which side has a larger value:

$$\sqrt{2} \ + \sqrt{5} \ + \sqrt{11}\$$ versus $$\ \ 7$$

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#### Jameson

Staff member
Interesting. Can you use all other buttons on your calculator? If there is a solution without using the calculator then I would like to see that very much. Guess, check and adjust isn't very elegant.

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#### checkittwice

##### Member
Interesting.
Can you use all other buttons on your calculator?

If there is a solution without using the calculator then I
would like to see that very much. Guess, check and adjust
isn't very elegant.
Only these keys** may be used:
-------------------------------

subtract

multiply

divide

parentheses

memory store

memory recall

equals/enter button

- - - - - - - - - - - - - - - - - - - - - - - -

And, you may use paper and something
with which to write on the paper.

<><><><><><><><><><><><><><><><><>

**This also includes, as an example, that you
cannot use an exponentiation key, such as
y^x. And then that eliminates the possible uses
of x^(1/2) and/or x^(0.5).

<> <> <> My solution may be forthcoming in a 1/2 day to 2 days
from now, so I could give users a chance.

#### Opalg

##### MHB Oldtimer
Staff member
Without using the square root button on a calculator,
determine which side has a larger value:

$$\sqrt{2} \ + \sqrt{5} \ + \sqrt{11}\$$ versus $$\ \ 7$$
In the inequality $\sqrt{2} + \sqrt{5} + \sqrt{11} \; \diamondsuit\; 7$, you have to decide whether the $\diamondsuit$ symbol should be < or >. Start by subtracting $\sqrt2$ from both sides: $\sqrt{5} + \sqrt{11} \ \diamondsuit\ 7 - \sqrt{2}.$ Now square both sides: $16 + 2\sqrt{55} \ \diamondsuit\ 51 - 14\sqrt2$.

Thus $2\sqrt{55} + 14\sqrt2 \ \diamondsuit\ 35$. Now square both sides again: $612 + 56\sqrt{110} \ \diamondsuit\ 1225$, and therefore $56\sqrt{110} \ \diamondsuit\ 613$.

So far, that has scarcely even needed a calculator. The last step is to square both sides again, and for that you do need the calculator, to get $344960 \ \diamondsuit\ 375769$, from which it is clear that $\diamondsuit$ has to be <.

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#### CaptainBlack

##### Well-known member
So far, that has scarcely even needed a calculator. The last step is to square both sides again, and for that you do need the calculator, to get $344960 \ \diamondsuit\ 375769$, from which it is clear that $\diamondsuit$ has to be <.
It might be convenient to use a calculator, but surely one does not need to use a calculator!?

CB

#### Opalg

##### MHB Oldtimer
Staff member
It might be convenient to use a calculator, but surely one does not need to use a calculator!?

CB
I suppose that depends on whether one remembers (or was ever taught) how to do long multiplication.

#### CaptainBlack

##### Well-known member
I suppose that depends on whether one remembers (or was ever taught) how to do long multiplication.
It is still taught (after a fashion) in UK junior schools (as of a few (<5) years ago) when my children were at that stage of their educations.

CB