Simplifying [1/(x+sqrt{x^2+1})][1+(2x)/(2sqrt{x^2+1})] and

[justinwager]

Hi....i'm new to the forum...but i'm not to sure how to post this...anyways...so i'm trying to figure out these two probelms and I have no idea how to start or even finish....lol but they're fairly difficult simplification questions...I don't know how to post them so I just scanned them...

1)

2)

Thanks for the help guys!

 

[mmm4444bot]

Use symbolic reasoning to make it easier ...

... i'm new to the forum Welcome!

... but i'm not to sure how to post this ... Click on Forum Help and read Karl's notes on typing math

... I have no idea how to start ... This statement is a Red Alert for a calculus student! :shock:

... but they're fairly difficult simplification questions ... No, actually, they are not. Use symbols to your advantage.

Hello Justin:

In order to do these exercises, you need lots of experience with adding, decomposing and simplifying fractions containing variables (rational expressions). You also need competency using the properties of exponents.

Some steps can be done in various orders, so there is more than one approach to simplifying these expressions. I will describe one approach.

Compare the two FACTORS in exercise 1. Do you see anything that appears the same in each?

Hopefully, you recognize that sqrt(x^2 + 1) appears in each.

Since this radical expression cannot be simplified itself, let's replace it with a new symbol everywhere it occurs, so that we can make exercise 1 more manageable.

\(\displaystyle a \;=\; \sqrt{x^2 + 1}\)

After replacements, exercise 1 looks like the following.

\(\displaystyle \frac{1}{x + a} \cdot \left(1 + \frac{2 \cdot x}{2 \cdot a}\right)\)

Next, do the addition inside the parentheses to get a single fraction, then multiply and simplify the product.

Finish by changing the symbol a back into sqrt(x^2 + 1).

Please let us know if you need help with any of these steps for exercise 1. Show results of your work so far, and explain why you're stuck.

The strategy for making exercise 2 more manageable is the same, but carry out the multiplication by the two factors of -1 in the numerator first.

Then, make replacements of single symbols for common expressions.

\(\displaystyle a \;=\; e^x + e^{-x}\)

\(\displaystyle b \;=\; e^x - e^{-x}\)

You should see the following.

\(\displaystyle \frac{a^2 - b^2}{a^2}\)

Decompose this rational expression into a difference of two fractions and simplify.

You should get 1 - (b/a)^2

Next, simplify b/a by going back to the corresponding expressions for a and b and applying to each the property of exponents that tells us e^(-x) = 1/e^x.

Do the addition of terms in a.

Do the subtraction of terms in b. For example, b becomes the following.

\(\displaystyle \frac{e^{2x} - 1}{e^x}\)

Now simplify the division of b/a by multiplying the single fraction b (above) by the reciprocal of the single fraction that you got for a.

Replace this result for b/a in 1 - (b/a)^2 and you're done.

If you need help with these steps for exercise 2, then please show the results of any work and explain why you're stuck.

Cheers,

~ Mark