While the radical symbol is widely used, converting to rational exponents has advantages. Explain an advantage
954754
It's what I have over you!
805031
1)  √10 · ³√100\n= 10^(1/2) · 10^(2/3)\n= 10^(7/6)\n\n2)  (6x)^(1/2) · (36x²)^(1/4)\n= (6x)^(1/2) · (6x)²^(1/4)\n= (6x)^(1/2) · (6x)^(2/4)\n= 6x\n\n...and so on.  Change the radicands to rational exponents with common bases, and follow the rules from there.  Good luck!
1005731
Here ya go buddy.  As x tends to infinity, only the highest power terms will dominate.  All you have to do is compare the highest power terms in under each root.  Effectively, the highest power of x in the first term is x^1 (since sqrt(x^2)=x) and the highest power in the second term is sqrt(x).\n\nNow, as x->inf, you are effectively looking at the function x-sqrt(x).  Since there are no denominator terms, this function goes straight to infinity.\n\nAlso, you could multiply by the conjugate to get \n(x^2+7x+50)/((sqrt(x^2+8x+50)+sqrt(x+2))\nbut you'd still have to go through the same kind of rationale, where you would be dealing with:\nlim x->inf  x^2/x = x \nwhich still goes to infinity.  Only use the conjugate if you can't make this kind of connection or if you are evaluating your limit at a finite value.
963683
there are 3 cases for a number A raised to the power of n:\n\n1) n greater than or equals to 1.\nWe simply say that "A is raised to the power of n"\n\n2) 0 < n < 1 and n is a rational fraction expressed as 1/m, m is an integer\nWe say that "A is raised to the mth root"\n\n3) n is any rational fraction not included in case 2, expressed as p/q, p and q are integers\nWe say that "A is raised to the power of p and qth root" which therefore involves both power and root\n\nAnd you're right about radicals being the sign used. By right, for square roots (which is actually 2nd root), we should add a "2" on the top left hand corner of the radical sign, but by default a radical without any number is taken as square root. But for all other nth roots, the radical sign must be accompanied by the number n.
1134236
Rational exponents are exponents that are rational numbers.\nThat is numbers that can be expressed as a fraction with whole number numerators and denominators.\n\nWhen a number raised to the 1/n power, it is the same as taking the  nth root of that number.  When a number is raised to the m/n power it is the same as taking the nth root and raising it to the mth power (or taking the mth power and then taking the nth root of that).\n\nA radical is the symbol that incicates that a root is being taken.
235721
(3x²)²(1/3y²)²=(9x^4)(1/9y^4)=(x^4)/(y^4)\n\nin many computer languages the "^" symbol is used for exponents.
1086154
The reason we call ln(x) "natural" is twofold: first, expressions in which the unknown variable appears as the exponent of e occur much more often than exponents of 10 (because of the "natural" properties of the exponential function which allow it to describe growth and decay behaviors), and second, because the natural logarithm can be defined quite easily using a simple integral or Taylor series--which is not true of other logarithms.
303643
Radical numbers are ones you are taking a root of.  The square root of 2, written as sqrt(2) (really written with that thing that looks like a long division sign).\n\nYou can take higher roots of numbers - cube root, for example, would look like a square root of a number, but with a 3 in the upper left of the outside of the sign (the 2 for a square root is omitted, and it is understood that if there is a square root sign with no number in that top left corner, then it is a square root).
258044
divide the exponent by two
317809
absolute values are important in higher levels of analysis when using wide ranges of numbers, also it help when we try to have equations of only real numbers make sense sometimes, i.e. various sorts of root equations
444064
So, for a) & b) the two things that you should notice are:\n1 - The exponents inside the second polynomial alternate, meaning that as the powers of the x variable decrease, the powers of the y variable increase, both by increments of one.\n2 - The answer will only contain the variables x and y raised to the same power, which is one more than the leading exponent on the x variable.  \n\nSince (x + y)(x^2 - xy + y^2) = x^3 + y^3\n and  (x + y)(x^4 - x^3y + x^2y^2 - xy^3 + y^4) = x^5 + y^5,\nthen for any given polynomial the pattern is written out...\n(x + y)(y^2n - x^(2n-1)y + x^(2n-2)y^2 + ... + x^2y^(2n-2) - xy^(2n-1) + y^2n = x^(2n+1) + y^(2n+1)
807333
Imaginary numbers, actually complex numbers, are enormously useful in many areas of engineering.  One example is in trigonometry where difficult trigonometric identities become almost trivial to derive.  Another is in analyzing circuits.  Using imaginary numbers allows Ohm's law to be applied to passive (resistors, capacitors and inductors) AC circuits.  Euler's identity allows cubic equations to be solved analytically.  Applying the concept of graphical representation of 2-D vectors provides very powerful tools for the solution of 2-D kinematics problems and difficult geometry problems.\n\nI am a research engineer and I use complex numbers all the time in my work.  It is by far one of the most useful tools that I have to solve difficult problems.
912338
Neither, it is undefined, i.e. it has none of the properties of all rational or whole numbers.\n\nI might be wrong, but I think this is one of those trick questions that are so common in math courses.
150881
There are several reasons. One of the most basic ones is that this representation provides a "polar" representation of complex numbers. Any complex number z=a+ib can be written as\n\nz= r exp(i theta)\nr= sqrt(a^2+b^2)\ntheta = atan(b/a)\n\nThis helps because, for example, a product of z1 and z2 would give\n\nz1 z2 = R exp(i phi)= r1*r2 exp (theta1 + theta2)\nR=r1*r2\nPhi = theta1 + theta2\n\nSo multiplying by z2 "stretches" z1 by amount r2 and "rotates" z1 by theta2. Such interpretations are convenient in many instances. Also, exponential functions are easier to deal with than regular a+ib type expressions in other cases.
1045033
Rational exponents are easier to combine when more than one appears in an expression.\n\nI would say that something like the following would be easier to solve with rational exponents:\nThe square root of x to the 5th power, all divided by the cube root of x to the fourth power.\nWhen I see this equation, the way to simplify is to change the radicals to rational exponents--x to the 5/2 power divided by x to the 4/3 power--which can be further reduced by subtracting exponents.\n\nSince my first step in that problem is to change the radicals to rational exponents, that seems to "prove" (in some sense) that rational exponents are easier to work with in this type of problem.\n\nSo my "short answer" is that rational exponents are much easier to work with when I am combining terms with multiplication and division.
233017
the number of roots for any equation is the degree of the equation. here it is 3 (coz the max power or exponent is 3)
400641
There's a difference.  It refers to (sin x)^2.  Plug in a few numbers and you can easily tell the difference.