What is the Value of a Complex Exponential Expression?

[sambarbarian]

Find the value of $\sqrt{-\sqrt{3}+\sqrt{3 + 8 \sqrt{7 + 4\sqrt{3}}}}$the options are $$1$$ , $$0$$ , $$2$$ , $$3$$

 

[sambarbarian]

sorry , i got so excited using latex for the first time i forgot to give my attempts

take the value as $$x$$

square both sides and take $$-sqrt{3}$$ to the other side , and continue doing till simplified , but this got way complicated than i intended.

 

[cepheid]

It's not clear what the problem is asking you to do here. The "value" of the expression is just what is given.

 

[sambarbarian]

oh , sorry again , the options are 1 , 0 , 2 and 3

 

[Saitama]

oh , sorry again , the options are 1 , 0 , 2 and 3

Start by writing 7+4√3 as 4+4√3+3 which can be simplified to (2+√3)^2.

 

[Mentallic]

Well just work backwards from the most inner radical and take approximations.

For the answer to be equal to 0, we need to have

$$\sqrt{-\sqrt{3}+\sqrt{3}}$$

which we clearly don't. For 1 we need

$$\sqrt{-\sqrt{3}+(1+\sqrt{3})}$$

And using the approximation of $\sqrt{3}\approx1.7$ would suffice.

For 2 we need

$$\sqrt{-\sqrt{3}+(4+\sqrt{3})}$$

And finally for 3 we need

$$\sqrt{-\sqrt{3}+(9+\sqrt{3})}$$

So what is the radical

$$\sqrt{3+8\sqrt{7+4\sqrt{3}}}$$ closest to? 2.7, 5.7 or 10.7?

 

[Ray Vickson]

Well just work backwards from the most inner radical and take approximations.

For the answer to be equal to 0, we need to have

$$\sqrt{-\sqrt{3}+\sqrt{3}}$$

which we clearly don't. For 1 we need

$$\sqrt{-\sqrt{3}+(1+\sqrt{3})}$$

And using the approximation of $\sqrt{3}\approx1.7$ would suffice.

For 2 we need

$$\sqrt{-\sqrt{3}+(4+\sqrt{3})}$$

And finally for 3 we need

$$\sqrt{-\sqrt{3}+(9+\sqrt{3})}$$

So what is the radical

$$\sqrt{3+8\sqrt{7+4\sqrt{3}}}$$ closest to? 2.7, 5.7 or 10.7?

Although I find it hard to believe, the original expression actually does come out exactly to a small integer value.

RGV

 

[Mentallic]

Although I find it hard to believe, the original expression actually does come out exactly to a small integer value.

RGV

The surds inside surds quickly lose their value!

What I find even more amazing is infinitely nested surds such as

$$\sqrt{10+\sqrt{10+\sqrt{10...}}}=\frac{1+\sqrt{41}}{2}\approx 3.7$$

Which is a lot smaller than you'd initially guess!

 

[ehild]

Start by writing 7+4√3 as 4+4√3+3 which can be simplified to (2+√3)^2.

Ingenious Pranav! And the same method can be applied again to get a small integer as result.

ehild

 

[SammyS]

Start by writing 7+4√3 as 4+4√3+3 which can be simplified to (2+√3)^2.

Ingenious Pranav! And the same method can be applied again to get a small integer as result.

ehild

Yes, Pranav-Arora !

I'm glad to see you figured it out before I saw this thread and racked my brain over this. (Of course, then I racked my brain over whether it's racked or wracked .)

 

[Saitama]

Thanks ehild and SammyS!

 

[sambarbarian]

Awesome solution pranav , can't believe i missed that. i got the answer 2 , thank you. btw which city are you from ?