What is Radical: Definition and 162 Discussions

In chemistry, a radical is an atom, molecule, or ion that has an unpaired valence electron.
A notable example of a radical is the hydroxyl radical (HO·), a molecule that has one unpaired electron on the oxygen atom. Two other examples are triplet oxygen and triplet carbene (꞉CH2) which have two unpaired electrons.
Radicals may be generated in a number of ways, but typical methods involve redox reactions. Ionizing radiation, heat, electrical discharges, and electrolysis are known to produce radicals. Radicals are intermediates in many chemical reactions, more so than is apparent from the balanced equations.
Radicals are important in combustion, atmospheric chemistry, polymerization, plasma chemistry, biochemistry, and many other chemical processes. A majority of natural products are generated by radical-generating enzymes. In living organisms, the radicals superoxide and nitric oxide and their reaction products regulate many processes, such as control of vascular tone and thus blood pressure. They also play a key role in the intermediary metabolism of various biological compounds. Such radicals can even be messengers in a process dubbed redox signaling. A radical may be trapped within a solvent cage or be otherwise bound.

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1. Help please with SI-ATRP (polymerization) of polyacrylamide

Can CuCl and Me6TREN form radicals? Reaction is at room temp and in water (not buffered)
2. Problems in manipulating these 4 radical equations

In order to write next step in all four equations above l used the definition of radicals. ##\sqrt a=b## means ##b^2=a##. Squaring both sides also works. I don’t know if it’s right. I mean I read that ##(\sqrt a)^2=a##. But I don’t know if we can apply this on expressions. Main problem is if we...
3. MHB 7.2.15 Int of trig in a radical

Evaluate the integral $I_4=\displaystyle\int_{-\pi}^{\pi}\sqrt{\frac{1+\cos{x}}{2}} \, dx$ ok offhand i think what is in the radical is trig identity but might be better way...

So I simplify the expression and get 2 x 3^2 x 7 = 84 and get ##2 \sqrt{21(y−2)^{3/2}}## I don't follow how the answer is ##2(y -2) \sqrt{21 ( y - 2) }##
5. I Is it possible to express cos(40) as a radical?

What sort of number is cos(40) ? You can solve the equation: $$4 \cos^{3}40 -3\cos40+0,5=0$$ but you end up with complex numbers requiring a cube root. The polar angle gets divided by 3 and you end up needing cos(20) or cos(10) in your answer. No way (it seems) to express as a radical or...
6. Solving radical equations: Do non-real extraneous solutions exist?

Assume you have a radial equation (eg. x+4 = √(x+10) ) that you want to solve for "x". To solve: \begin{align} (x+4 & = \sqrt[2] {x+10})^2 \nonumber \\ (x+4)^2 &= |x+10| \nonumber \end{align} For my question, we are only going to consider the case where x+10 < 0: \begin{align}...
7. I Simplifying a nested radical that includes a complex number

Dear Everyone, This post is not a homework assignment... I want to use the quartic formula. In one step is to solve the resolvent cubic. I know that there is 3 real solutions this particular resolvent cubic. I want to know how Bombelli got his answers before the discovery of the trigonometric...
8. MHB Finding the Value of a Trigonometric Integral with Radical

Evaluate $$\int_0^{2\pi}\sqrt{\dfrac{1-\cos{x}}{2}}\,dx$$ ok my baby step is $$\int _0^{2\pi }\frac{\sqrt{1-\cos \left(x\right)}}{\sqrt{2}}dx$$ then ? W|A said the answer was 4
9. MHB 7.2.1 integrate trig with radical

Ok I tried to follow an example but kinda got lost in this maybe more steps the answer is from W|A
10. MHB Inequality involves radical, square and factorial expression 3√{x}+2y+1z^2⩽ 13

If $x^2+y^2+z^2+xyz=4$ and that $x,\,y,\,x\ge 0$, prove $3!\sqrt{x}+2!y+1!z^2\le 13$.
11. Big integer arithmetic functions

NOTE:This is not a homework question! This is just a topic that I like very much,but don’t have the programming ability to do many of them.That’s why I post this thread. C++ is a language without built-in big integer calculation functions,so building ones that can do such job is a great way to...
12. B Bounds for the index of a radical

Hi, I know that, at least formally, the index of a radical should be positive and integer. That is if I introduce \sqrt[x]{2} I need to assume x\in \mathbb N and x>0. However, my calculator has no problem in calculating the radical for any x\neq 0, say x=-\pi. The result it gives is based on...
13. Finding limits with a radical in denominator

Homework Statement Evaluate: ##\lim_{x \rightarrow -\infty} {\frac{3x^3+2}{\sqrt{x^4-2}}}## Homework EquationsThe Attempt at a Solution For limits involving fractions, it's a good idea to divide the numerator and the denominator by the highest degree x in the fraction. In doing this, we can...

I am seeing from my results that an oxidation process is taking place, the pollutants degradation correlates closely with the formation of radicals. However, in one case there are less radicals produced and there is no degradation... I would like to ask if on occasions there must be a...

Homework Statement For the solution to a given problem, in the second to last step I had: ##-\frac{\sqrt 6}{4} + \frac{\sqrt 2}{4}## I stated next that the solution was ##-\frac{\sqrt{6}+\sqrt{2}}{4}## I was told this was incorrect and that the correct solution is...
16. I'm having some trouble differentiating radical functions

Homework Statement Find the derivative of the following functions. h(x)=8x2√x2+1 Homework EquationsThe Attempt at a Solution I just had a lesson on friday but it was a bit confusing to me. I am able to solve some problems but ones that contain multiple rules confuse me. If someone could...
17. MHB Algebraic Verification of Radical Equations: Solving Without a Calculator

Verify that both sides of the radical equation agree without using a calculator. See picture. How can this be done algebraically?
18. MHB How to Verify a Radical Equation Algebraically?

Verify that both sides of the radical equation agree without using a calculator. See picture. How can this be done algebraically?
19. MHB How can a radical equation be verified algebraically without a calculator?

Verify that both sides of the radical equation agree without using a calculator. See picture. How can this be done algebraically?
20. MHB Solving radical equations x√3+x√2=1

I'm trying to help my son with homework and can't get the following problem x\sqrt{3}+x\sqrt{2}=1 Please help
21. MHB Is FOIL used to multiply radicals in sqrt{x - 3} • sqrt{x}?

The picture shows a simple problem. However, my question has to do with multiplication of radicals. I know how to use FOIL. sqrt{x - 3} • sqrt{x} is slightly confusing. Do I multiply radicand times radicand? My question is: Does sqrt{x - 3} • sqrt{x} become sqrt{x^2 - 3x} in the FOIL...
22. MHB How can we solve a radical equation without using a calculator?

Show that the RHS = LHS without using a calculator.
23. MHB Can the Algebraic Expression $8-2\sqrt7$ be Simplified to a Perfect Square?

See picture. Show that the RHS = LHS without using a calculator.
24. MHB Integrating the Square Root of a Fraction with a Radical

$\tiny{10.08.06}\\$ $\textsf{Evaluate the function}$ \begin{align*}\displaystyle I_5&=\int \sqrt{\frac{x^2-4}{x}} \, dx \end{align*} ok, I thought this would be a simple U subst, but nothing looks convienent

When solving radical equations, we must check for extraneous roots. What are extraneous roots?
26. MHB Solve Radical Equation: 5x√2x-3=4

solving this problem not sure how to calculate this 5x√2x-3=4 i was planning to isolate radical by subtracting the 5x when i do this i end with a quadratic and no solutions any help appreciated
27. MHB How do you simplify sqrt{12x^7}?

Simplify sqrt{12x^7} Solution: sqrt{4•3•x^6•x} 2x^3(sqrt{3x}) Is this correct? Note 1: According to wolfram, the answer is 2sqrt{3}•sqrt{x^7} Note 2: The term x^3 in my answer comes from the breakdown of sqrt{x^7} as sqrt{x^6•x}. Isn't sqrt{x^6} the same as x^3?

(-2sqrt{8 - 2{7}})^2 (-2sqrt{8 - 2{7}})(-2sqrt{8 - 2{7}}) 4(8 - 2sqrt{7}) Correct?
29. MHB Radical Equation Without Constant

Solve for all real number t values. sqrt {2t + 5} - sqrt {8t + 25} + sqrt {2t + 8} = 0 I see there are no constants in this problem. I typically isolate the radical on one side of the equation and the constant (s) on the other side but there are 3 radicals on the left side. This is strange...
30. MHB How can I solve radical equations using the given hint?

Determine all of the real number solutions for the radical equation. sqrt {x^4 - 13x^2 + 37} = 1 Hint given in textbook: Let x^2 = t and x^4 = t^2 After applying the hint given, do I proceed as usual by squaring both sides? I have to back-substitute somewhere in the solution steps, right...
31. MHB How to Simplify a Radicand by Breaking it into Two Parts?

How does (2sqrt{7})(sqrt{8 - 2sqrt{7}) become 2(7 - sqrt{7})?
32. MHB How to Solve the Radical Equation Using Fifth Root and Square Root?

Let frt = fifth root Let sqrt = square root Show that frt{176 + 80sqrt{5}} = 1 + sqrt{5} Do I raise both sides to the 5th power? Can someone get me started?
33. MHB Can Cubic Roots and Square Roots Combine to Equal One?

Let cbrt = cube root Let sqrt = square root Show that cbrt{2 + sqrt{5}} + cbrt{2 - sqrt{5}} = 1 without using a calculator. Can someone get me started? Do I raise both sides to the third power as step 1?
34. MHB Is it possible to simplify this radical equation without using a calculator?

Show that the left side equals the right side without a calculator. sqrt{7} - sqrt{8 - 2sqrt{7}} = 1 I know squaring must be done here and probably more than once. I am stuck in terms of squaring the left side. [sqrt{7} - sqrt{8 - 2sqrt{7}}]^2 = (1)^2 Can someone square the left side for me...
35. Finding the derivative of a function with a radical

Homework Statement Find the differential Homework Equations Chain rule : dy/du=dy/du*du/dx Product rule: f(x)g'(x) + g(x)f'(x) The Attempt at a Solution I have tried to move the radical to the top of the equation by making it into an exponent (x^2+1)^-1/2. I then used the product rule and the...
36. I Jacobson Radical and Right Annihilator Ideals ....

I am reading Paul E. Bland's book, "Rings and Their Modules". I am focused on Section 6.1 The Jacobson Radical ... ... I need help with the proof of Proposition 6.1.7 ...Proposition 6.1.7 and its proof read as follows: In the above text from Bland ... in the proof of (1) we read the...
37. MHB Jacobson Radical and Direct Sums - Bland, Proposition 6.1.4 ....

I am reading Paul E. Bland's book, "Rings and Their Modules". I am focused on Section 6.1 The Jacobson Radical ... ... I need help with the proof of Proposition 6.1.4 ... Proposition 6.1.4 and its proof read as follows: In the above proof from Bland we read:"... ... If i_1 \ : \ M_1...
38. I Jacobson Radical and Direct Sums - Bland, Proposition 6.1.4

I am reading Paul E. Bland's book, "Rings and Their Modules". I am focused on Section 6.1 The Jacobson Radical ... ... I need help with the proof of Proposition 6.1.4 ...Proposition 6.1.4 and its proof read as follows: In the above proof from Bland we read:"... ... If ##i_1 \ : \ M_1...
39. What's the Error in Solving Radical Equations?

Homework Statement √3x - 5 +2 = -3 Underlined is under square root. √2x - 3 = -x + 3 Underlined is under square root. Homework EquationsThe Attempt at a Solution √3x - 5 +2 = -3 -2 -2 (√3x - 5)2 = (-5)2 3x - 5 = 25 3x = 30 x = 10 Solution says no answer but I got one... I tried...

Homework Statement (Square root)-12 -3x - 3 = 0 Everything underline is supposed to be under the square root sign. Homework EquationsThe Attempt at a Solution (Square root3x +12)2 = (3)2 3x + 12 = 9 -12 -12 3x = -3 x = -1 Solution in this learning guide says the answer is -7. So I'm...

I am reading Paul E. Bland's book, "Rings and Their Modules". I am focused on Section 6.1 The Jacobson Radical ... ... I need help with the proof of Corollary 6.1.3 ... Corollary 6.1.3 (including the preceding Proposition) reads as follows: My questions are as follows: Question 1 In the...

I am reading Paul E. Bland's book, "Rings and Their Modules". I am focused on Section 6.1 The Jacobson Radical ... ... I need help with the proof of Corollary 6.1.3 ...Corollary 6.1.3 (including the preceding Proposition) reads as follows: My questions are as follows:Question 1 In the proof...
43. B Solving equations with radicals (extraneous solutions)

I am solving the equation ##\sqrt{x + 3} + 4 = \sqrt{8x + 1}##. I understand that , generally, to solve it, we have to eliminate the radicals by isolating a radical expression to one side and then squaring both sides of the equation. I end up obtaining two solutions: ##x = 6## and ##x =...
44. B What use is there for rationalizing denominator?

I've been learning algebra for the past 2 years (in high school), not once have we ever had to rationalize a denominator in a radical expression. I am now relearning Algebra and Trig., what use is there? I mean, all you're doing is switching the numerator (rational) to the Denominator...
45. Find arclength of curve; stuck trying to integrate radical

Homework Statement Find the arclength of the curve x = ⅓(y2+2)3/2 from y=0 to y=1. Homework Equations (Please forgive the crazy definite integral symbols - I'm taking a LaTex class tomorrow so hopefully I'll be able to communicate more clearly from then on..!) arclength of curve = ∫ab √ (1 +...
46. Free Radical Stability Homework: Find the Optimal Temp

Homework Statement Homework EquationsThe Attempt at a Solution option 1: 1 degree C option 2: 1 degree C option 3: 3 degree C option 4: 1 degree C SO the answer should be option 3?
47. Hydroxide Radical: Exploring Alkaline Properties of Water

Dear PF Forum, Recently I'm studying an ionizer water machine. This machine claims (it's what it says) that it can turn 'ordinary' water into alkaline water and good for health. It's not the machine that I ask, but it's the alkaline property that I want to know. Why is the walter alkalinized...
48. NBS vs radical HBr in the bromination of olefins

I am having some trouble understanding why different sources of bromine radicals supposedly brominate an alkene at different positions. What I mean by this: In the first example, the Bromine radical attacks a hydrogen at the allylic position and then a termination reaction results in a Bromine...
49. B The Problem That Riddles Any Radical Thinker: WTH Is Energy

Hello! I'm new to this forum but ok let's get this started, First I'll revisit the definition I know as physics; the study of nature in order to provide a reasonable(radical/rudimentary) explanation/reason. This I believe in is not quite how things are today, at school, with my teachers or at...
50. MHB Radicals and Radical Ideals, Dummit and Foote Proposition 11, Section 15.2

I am trying to gain an understanding of the basics of elementary algebraic geometry and am reading Dummit and Foote Chapter 15: Commutative Rings and Algebraic Geometry ... At present I am focused on Section 15.2 Radicals and Affine Varieties ... ... I need help on an apparently simple...