How Do You Simplify 4√(sin(π/4))?

[2h2o]

I've come across this before, but for the life of me can't seem to get anywhere with it. I can plug this into technology and get an answer (and then move on...), but I want to know how to do it without.

An example problem:

$$4\sqrt{\sin(\pi/4)}$$

I get as far as

$$2\sqrt{2\sqrt{2}} = 2(2(2)^\frac{1}{2})^\frac{1}{2}$$

but then I run out of steam and don't see the next step. Surely it is something simple that I am missing, but I just don't see it.

Thanks for any tips. Cheers!

 

[Char. Limit]

I don't know what you're asking, but since $$2\sqrt{2}=\sqrt{8}$$, it equals two times the quartic root of 8...

 

[2h2o]

I have $$\sqrt{8}$$ as an intermediate step in my work.

$$4\sqrt{\sin(\pi/4)} = \sqrt{\frac{16\sqrt{2}}{2}} = \sqrt{8\sqrt{2}} = \sqrt{4*2\sqrt{2}} = 2\sqrt{2\sqrt{2}}$$

When I plug into technology, I get the answer:

$$2*2^\frac{3}{4}$$

but I don't know how to get there. That's what I'm asking; sorry for being unclear.

 

[Char. Limit]

Well, yeah.

Going with my two times quartic root of 8 solution, since 8 is 2 cubed, the quartic root of 8 is 2 to the three-fourths power, and the number simplifies to your number.

 

[2h2o]

Of course. Thank you very much! I definitely should've seen that! :)

Cheers.

 

[jacksonwalter]

I've come across this before, but for the life of me can't seem to get anywhere with it. I can plug this into technology and get an answer (and then move on...), but I want to know how to do it without.

An example problem:

$$4\sqrt{\sin(\pi/4)}$$

I get as far as

$$2\sqrt{2\sqrt{2}} = 2(2(2)^\frac{1}{2})^\frac{1}{2}$$

but then I run out of steam and don't see the next step. Surely it is something simple that I am missing, but I just don't see it.

Thanks for any tips. Cheers!

When you multiply a two numbers with the same base, the exponents add. So,

$$2(2^{1}*2^{\frac{1}{2}})^{\frac{1}{2}} = 2(2^{1+\frac{1}{2}})^{\frac{1}{2}} = 2(2^{\frac{3}{2}*\frac{1}{2}}) = 2*2^{\frac{3}{4}}$$