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Begin by dividing both sides of the equation by sqrt(y). This gives, ( dy /dx ) / sqrt(y) = sqrt(x), and after multiplying by dx, dy / sqrt(y) = sqrt(x) dx. Now integrate both sides and get S dy/sqrt(y) = S sqrt(x) dx ... 2*sqrt(y) = (2/3)*x*sqrt(x) + C. The initial...

2π r2 + 2π r(3r) = 2π r2 + 6π r2 = 8π r2. I hope this helped, and if you need more assistance, I or another tutor will be happy to help.

z3 + 125 = z3 + 53 = (z + 5)(z2 - 5z + 25). This follows from the "sum of cubes" formula, below. a3+ b3 = (a + b)(a2 - ab + b2). Hope this helps. If there's any more I can do to help please ask. I or another tutor...

Any choice of points on a line will give you the same result when you calculate the slope. For lines slope is a constant, and so will not be dependent on which points you choose to "plug into" the formula. Hope this answers your question, and if there is anything else you need assistance...

There is a formula well-known as the point-slope equation. It gives the equation of the line through a given point with a given slope. Here it is: y - y1 = m(x - x1), where (x1, y1) is the given point and m is the slope. I hope this helps. If you would like more assistance, please feel...

-9x - 13 - 9x is equivalent to -9x - 13 + (-9x). Remember that subtracting a number is equivalent to adding its opposite. Put another way, a - b = a + (-b). I hope this helps, and if you need more explained, please feel free to ask. I or another tutor will be happy to assist...

I'm reading your equation as ( 2x + 5 )/5 = ( x - 2 )/3. "Cross-multiplying" refers to rewriting 'A/B = C/D' as 'AD = BC.' Try this first.

Here it looks like they would like expressions written algebraically that are equivalent to the given statements involving the variable 'n.' An few examples might clear it up for you. (i) two more than a number n.   n+2. (ii) four times a number n.  4n. (iii)...

Problem:  To maximize  8R + 10F [profit] subject to the two constraints 8R + 7F <= 208 [fab dept], R + 3F <= 60 [fin dept]. Work with this and if you would like more help, I or another tutor will be happy to assist you.     &...

The Alternating Series Test ∑(-1)nBn converges when the following two conditions are met: (i) lim Bn = 0 and (ii) {Bn} is (eventually) decreasing. Note: AST doesn't apply when either of the conditions is not met, and so never is a test for divergence.  If the first condition isn't...

Remember the following rules when dealing with exponents. am*an = am+n. am/an = am-n. a-n = 1/an. Also remember that a0 = 1. Also it may help to remember that you can regroup the terms according to bases. To illustrate I'll use another example, and leave yours for...

Sadly, I'd say there are no famous mathematicians today, in the same sense that celebrity athletes, actors, politicians, and the like are famous. But that doesn't stop people from going into the field and continuing the work of mathematicians past.  They just don't do it for the fame...

Read the instructions for this problem carefully, and make sure you have included them in this post. As you've written it here so far, I can't figure what the problem is asking you to do. I certainy could guess, but that wouldn't be too instructive, and I would really like to help.

Yes they are the same number, but LCM, for Least Common Multiple is a more descriptive term, as it hints at a way to find it. To find these, one could list the multiples of each number (or denominator). For example, say one needs the LCM of 6 and 15. 6:  6, 12, 18, 24, 30, 36,...

I think a technique called "completing the square" is needed here.  It rewrites y = Ax2 + Bx + C in the form y = A(x + P)2 + Q. It is accomplished in several steps.  To demonstrate, I'll use an example like your own, leaving it for you to try yourself...

Perhaps a Truth Table might shed some light on this.  Below is a TT for "if p, then q." p. q. if p, then q. T.  T.  T. T.  F.  F.  [note this case.  "if T, then F" = F.] F.  T.  T. F.  F.  T. Notice...

The "multiples" of a number, N, are the numbers N, 2*N, 3*N, ..., etc.  For example, the multiples of 4 are listed below in the usual order. 4, 8, 12, 16, 20, ..., and so on. So by "common multiple" one means a number that is a multiple of 2 or more numbers. Try...

(a + b)2 = (a + b)(a + b) = a2 + ab + ba + b2 = a2 + 2ab + b2. Above I've given a short derivation of a very useful formula which demonstrates how to "square a binomial," which is just a fancy way of saying "raise the sum to the 2nd power." In...

First work out the area by multiplying the dimensions of the fence. Now that you have the area, multiply this by the unit cost per square foot, to get the pre-tax cost of the fence. Now calculate the tax by multiplying the pre-tax cost, by 7% or .07.   After adding the tax to the...

Let's work an example like yours that will require similar steps to complete. Say we need to solve the following equation for 'b': (5-b) + 4(3b - 1) = 23. The first step would be to expand the LHS by distributing the 4. 5 - b + 12b - 4 = 23. Now simplify this as follows,...