Simplifying Square Roots

[FNP_sr14]

So I was wondering how you would simplify 2a squared + 1 all under the square root sign? I know that 2a squared simplifies to "a" times the square root of 2, but I'm not quite sure what to do with the 1. Help?

 

[tkhunny]

Very bad. You do not know that because it is not the case.

\(\displaystyle \sqrt{2a}\;=\;\sqrt{2}\cdot\sqrt{a}\)

This is reasonable, albeit not particularly useful.

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

Unless you can find a common FACTOR in those two TERMS, just leave it alone.

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

Common FACTORS.

 

[Mrspi]

So I was wondering how you would simplify 2a squared + 1 all under the square root sign? I know that 2a squared simplifies to "a" times the square root of 2, but I'm not quite sure what to do with the 1. Help?

sqrt(2a[sup:b15k3j8a]2[/sup:b15k3j8a]) DOES, in fact, simplify to |a| sqrt(2), because a[sup:b15k3j8a]2[/sup:b15k3j8a] is a factor of 2a[sup:b15k3j8a]2[/sup:b15k3j8a]...the absolute value symbol is needed because a square root is understood to be positive.

But, if what you have is sqrt(2a[sup:b15k3j8a]2[/sup:b15k3j8a] + 1), then you have to look at whether the expression under the radical sign has any perfect square factors.. In this case, the only factor of 2a[sup:b15k3j8a]2[/sup:b15k3j8a] + 1 is 1. And removing a factor of sqrt(1) doesn't help simplify anything.

Sometimes, a radical cannot be simplified. In fact, in the grand scheme of things, I'd guess it is more likely than not that a random radical can't be simplified.

 

[mmm4444bot]

2a squared + 1 all under the square root sign

I read this phrase as: sqrt([2a]^2 + 1)

 

[Mrspi]

Re:

2a squared + 1 all under the square root sign

I read this phrase as: sqrt([2a]^2 + 1)

Fair enough....it would make our job considerably easier if we did not have to guess at what is actually meant by the problem as posted. 2a[sup:1xiqcxrl]2[/sup:1xiqcxrl] and (2a)[sup:1xiqcxrl]2[/sup:1xiqcxrl] surely don't represent the same algebraic expression....but regardless, the expression in the original post doesn't have any perfect square factor other than one.

 

[mmm4444bot]

it would make our job considerably easier if we did not have to guess

I hear you.

Putting into the Read Before Posting message a few examples of basic text-notation for math is one of the suggestions that I'm thinking of.