In chemistry, a radical is an atom, molecule, or ion that has an unpaired valence electron.
With some exceptions, these unpaired electrons make radicals highly chemically reactive. Many radicals spontaneously dimerize. Most organic radicals have short lifetimes.
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.
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...
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) }##
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...
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}...
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...
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
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...
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...
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...
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...
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...
$\tiny{10.08.06}\\$
$\textsf{Evaluate the function}$
\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
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
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?
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...
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...
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?
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?
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...
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...
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...
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...
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...
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...
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 =...
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...
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 +...
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?
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...
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...
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...
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...