It's not "radical Islam". It's Islam.

To say "radical Islam" is like saying that the Pope is a "radical Catholic". The word "radical" connotes sweeping or revolutionary change, and as a synonym of "extremism" usually is only apt as a modifier of an action or actor seeking to create change. Muslim terrorists are not trying to change Islam.

Spencer's The Complete Infidel's Guide to the Koran You could call it "fundamentalist" or "orthodox Islam" -- but save for some tiny obscure little sects here and there that have mixed up Islam with tribalism and other religions, there are no actual ideologies of "moderate" or "reform" Islam, so neither of these terms is entirely accurate either.

You could say "extremist Muslim" but again, that's like calling the Pope an "extremist Catholic", or a rabbi an "extremist Jew".

You could call them "Muslim jihadists" or talk about jihadi, but that ignores that the expression of the faithful's struggle that is called "jihad" (which applies to "spiritual" things as well as pugilism) is integral to Islam, so that's like trying to identify fanatic Christians by saying that they are believers in Christ.

You could call them "violent jihadists", but you do so at your own risk, since violent overthrow and submission of non-Muslim cultures and nations is but only one form of jihad, and it's repeatedly encouraged in later-written sections of the Koran. (Unlike the Bible, later verses of the Koran supersede earlier verses, so don't be a credulous patsy and fall for the dissimulation pointing to this or that isolated earlier verse in the Koran to make the specious argument that Islam is a "religion of peace").

Islam: the religion of peace -- whatever 
There is something fundamentally wrong with a religion that through the centuries even into modern times consistently has attacked others, from the days Mohammed forcibly took over Medina and killed the Jews who lived there who wouldn't submit to him (yes, it was a Jewish community first), to the invasions of Europe that gave rise in turn to the Crusades, to the Barbary pirates, to the sympathizing with the Nazis in WWII, to the continuous insane 10 or 20 percent of violent, suicidal lunatics who blow up themselves and others thinking they are doing God's will.

The only "peaceful" or "moderate" Muslims are those who just don't pay all that much attention to their religion, who are culturally Muslim but otherwise basically non-believing and non-practicing, or practicing only superficial cherry-picked bits, such as a holiday here and there. Most of these people, however, still share a kinship, loyalty and identification with Islam, the warm nostalgic fuzzies inculcated in childhood. And that's alarming when one considers human nature and the repeating history of tyranny in the world. It takes only a small percentage of the committed, and then the mildly sympathetic or previously indifferent majority just goes along with the program. (For example, only seven percent of Germans pre-WWII were Nazis.)

If someone wants to believe in their "religion", that's all good and well if acting on those beliefs doesn't cause harm to anyone else. But that -- and only that -- is what the First Amendment freedom of religion is about. In constitutional law "your right to swing your arm ends at the tip of my nose."

Islam is not merely a religion, however. It is a political ideology. That political ideology is incompatible with, and stands for the overthrow, violent or otherwise, of the entire legal and philosophical foundations of every western society and government. In this regard, it is the world's largest hate cult. But because this political system and these beliefs are entwined with religion, Islam, as all religions, inevitably and continuously will give rise to some percentage of the inculcated who are "very devout". In Islam, though, "devout" becomes something dangerously much more than the Catholic devotee who goes to church every day, or the Jainist who rescues errant worms in the soil when he harvests his carrots, or the Orthodox Jew who wears medieval-like attire, or the Christian fanatic who preaches about the end of the world on the street corner.

Given this reality, the only appropriate response to those who disavow that they adhere to the political blatherings of the megalomaniac who made up stories about "Allah", who do not yell (or think) "Death to America", and who claim that they do not belong to or support any organization whose members bomb, stone, behead, burn, knife, hang, whip, and otherwise murder other people is: "PROVE IT".

Stop carrying on in defense of this nonsense parading as a "belief system". Stop exalting it, stop excusing it, stop lying about it, stop taking pride in its symbols, and just renounce Islam.

We don't put up with people claiming to be "peaceful" or "moderate" members of the KKK, or of the Nazi party, etc. So no thinking person should swallow these very same claims vis a vis Islam just because its founder justified his power to credulous primitives by claiming that he had received edicts from God.