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Method #1 is when you are multiplying two decimal numbers, count how many digits are after the decimal point for each number. For example, (36.58)(2.4), there are two digits after the decimal point in 36.58 and one digit for 2.4. Add these to get three, so after you multiply, move the decimal point three positions to the left to compensate.

Method #2 is intuition (because 36.58 times 2.4 can't equal 877.92 can it?).

There's really only one method (and maybe variations) to find where to put the decimal point, though.

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Not enough information.

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{9,10,11},{10,11,12},{11,12,13},{12,13,14}. ({13,14,15} doesn't work)

Just eyeball it, roughly approximate by dividing by 3 and you can find all such sets.

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You want the next three numbers, not the next three sets of numbers. Even then, your sequence is not very well defined.

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You can find the roots of x by applying by the quadratic formula, but you can't solve for a numerical value of x unless you set f(x) equal to something (e.g. 0).

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You can't "solve" it as you might solve an equation. You can graph it or find its derivative or something but you can't "solve" tan(x)/x.

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Factor LHS

x(a+b) = c, divide both sides by a+b.

x = c/(a+b)

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For any binary operation * (not necessarily addition or multiplication):

* is said to be associative if (a*b)*c = a*(b*c) for all a,b,c ∈ S (S is a set composed of all possible a,b,c)

* is said to be commutative if a*b = b*a for all a,b ∈ S

* is said to be distributive over another operation (I'll denote this operation by &) if a*(b&c) = (a*b)&(a*c) for all a,b,c ∈ S.

For example, vector addition is commutative and associative, scalar products are distributive, the dot product of vectors is commutative, and the cross product of vectors is associative over vector addition, but not commutative (in fact, it is said to be anti-commutative, similar to subtraction). These are just examples of operations other than addition or multiplication that obey these properties.

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Assume the radius of the cylinder is r. Upon drawing the sphere and cylinder, draw a right triangle (with a vertex at the center of the sphere) with legs r and h/2, and hypotenuse 10 (since the distance from the center to the edge of the sphere is 10). This will satisfy the Pythagorean theorem







Now use the volume and surface area formulas:





and replace your expression for r into each of these.

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a. Do the operations in parentheses first...e.g. 9-5 = 4, 6-8 = -2. Then (9-5)-(6-8) = 4-(-2) = 4+2 = 6.

b. -8 is the smallest of the four numbers.

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You can't "do" a certain arithmetic sequence; you have to explain what "do" means.

If a sequence is denoted a_1, a_2, ..., a_n, ... and n = 19, then a_19 denotes the 19th term of the sequence. d=8 is the common difference; i.e. each term in the sequence is 8 more than the previous one. a_n = 155 means the nth term is 155. You'll have to rephrase your problem for it to make more sense and obtain an answer.

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You want to find the arithmetic mean of one 1, two 2's, ..., five 5's, which is essentially



since the sum of the entries is 1^2 + ... + 5^2 and there are 1+2+...+5 entries.

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What is the extra 2 doing there? Hopefully the height is not 4 meters in base 2, which would be meaningless and incorrect.

Even then, what would you want to do with such a rectangle?

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What point?

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Same units digits as 7^n. 7^1 ≡ 7 (mod 10), 7^2 ≡ 9, 7^3 ≡ 3, 7^4 ≡ 1, 7^5 ≡ 7, and the pattern repeats. The only possible digits are 1,3,7,9.

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The z-score would be 2, since it is two standard deviations above the mean.

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No guessing...to find the x-intercept, set y equal to zero and solve for x. Similarly, to find the y-intercept, set x equal to zero and solve for y.

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1. There are none. Since x < y < z, the maximum value of the LHS is about 1 (set x = 1, y,z to be large numbers). The LHS can never get larger than 1, so there are no solutions (x,y,z) of positive integers.

2. Out of any five integers, there will either exist an integer for each residue modulo 3 (in which we're done), or all five integers will occupy two different residues mod 3. If this is the case, then by a simple Pigeonhole argument, three of them have the same residue mod 3, in which we're also done (since the sum of these three is 0 mod 3).

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b+4

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18(-18) = -324

30(-16) = -480

18(-18) + 30(-16) = -324 + (-480) = -804

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Sum of 2819? Addition is a binary operation, so you need two numbers.

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You didn't specify which "sets." Assuming you mean the "double-stroke" sets i.e. real, rational, integer, whole number, natural number, 1.93 is a real number and a rational number (193/100) but not an integer.

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Either complete the square or factor. If neither of these work very well, you can use the quadratic formula. But you only need to use one method (as long as you do your arithmetic correctly). As to eliminating/adding steps, there is no need to do so, and no need to change the steps. I can't really answer those questions; you'll learn much better by practicing solving quadratic equations.

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If the number that the variable represents satisfies the equation (i.e. you get an equality or true equation).

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5 units by 6 units. You can do this by noting that 30 = 5*6, or by doing it the longer way by letting x(x+1) = 30 and solving via factoring.

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Just plug in the variable that is known and solve for the other number. For example,

"what is y if x is 8?"

Just replace x = 8 to get y = (-3/4)(8) - 6 = -12.

Do the others the same way.

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sin(5pi/6) = 1/2. Just find 5pi/6 on a unit circle and find the y-coordinate (using 30-60-90 triangles).

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Is 32 equal to -4?

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Just subtract that number from 180 to get 180 - (2x - 4) = 180 - 2x + 4 = 184 - 2x.

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There are lots of counterexamples, so you do not need to refer to it as "the counterexample."


For example, a 4 ft * 6 ft rectangle and a 3 ft * 7 ft rectangle both have perimeter 20 ft, but their areas are different (24 ft^2 vs. 21 ft^2).