# What is the clue to think that way?

• Jan 3rd 2011, 11:08 AM
SWEngineer
What is the clue to think that way?
In the "Foundation Maths" book in chapter (19): "The exponential function", there was an example solving the following problem (I used "^" to denote power):

(e^2t + 2e^2 + 1)^1/2

In solving the problem, this was converted to ((e^t + 1)^2)^1/2

What clue can tell that I have to solve the problem this way and not by just taking the root of every element?

Thanks.
• Jan 3rd 2011, 11:28 AM
Quote:

Originally Posted by SWEngineer
In the "Foundation Maths" book in chapter (19): "The exponential function", there was an example solving the following problem (I used "^" to denote power):

(e^2t + 2e^2 + 1)^1/2

In solving the problem, this was converted to ((e^t + 1)^2)^1/2

What clue can tell that I have to solve the problem this way and not by just taking the root of every element?

Thanks.

$\displaystyle \sqrt{e^{2t}+2e^2+1}=\sqrt{\left(e^t+1\right)\left (e^t+1\right)}$

Taking the square root of each term does not work.

$\displaystyle \sqrt{4}+\sqrt{4}=2+2=4$

$\displaystyle \sqrt{4+4}=\sqrt{8}=\sqrt{(4)2}=\sqrt{4}\sqrt{2}=2 \sqrt{2}<4$

Or

$\displaystyle \sqrt{8}<\sqrt{9}\Rightarrow\ \sqrt{8}<3<4$

To take a square root, you can do so when you are taking the square root of a square.

$\displaystyle 2^2=4\Rightarrow\sqrt{4}=2$

$\displaystyle 3^2=9\Rightarrow\sqrt{9}=3$

$\displaystyle 3^2+4^2=5^2\Rightarrow\sqrt{3^2+4^2}=\sqrt{5^2}=5$

$\displaystyle \left(e^t+1\right)\left(e^t+1\right)=e^t\left(e^t+ 1\right)+1\left(e^t+1\right)=e^{2t}+e^t+e^t+1=e^{2 t}+2e^t+1$

However, this is a square

$\displaystyle e^{2t}+2e^t+1=\left(e^t+1\right)^2$
• Jan 3rd 2011, 12:09 PM
HallsofIvy
Quote:

Originally Posted by SWEngineer
In the "Foundation Maths" book in chapter (19): "The exponential function", there was an example solving the following problem (I used "^" to denote power):

(e^2t + 2e^2 + 1)^1/2

In solving the problem, this was converted to ((e^t + 1)^2)^1/2

What clue can tell that I have to solve the problem this way and not by just taking the root of every element?

Thanks.

The "clue" is recognizing perfect squares. x^2+ 2x+ 1= (x+ 1)^2

Here, x= e^t so x^2= (e^t)^2= e^{2t}
• Jan 4th 2011, 01:47 AM
SWEngineer
Thanks a lot for your replies. It is clear now.