TwentyFour

Hint:

You're trying to find \(z^{53}+z^{52}+z^{51}+z^{50}+z^{49}\), so factor out \(z^{49}\) to get \(z^{49}(z^4+z^3+z^2+z+1)\). We know what \(z^2+z+1\) is, so substitute it in. You now have\(z^{49}(z^4+z^3)\), which can be written as \(z^{51}(z^2+z)\). Now, we know that \(z^2 + z + 1=0\), so \(z^2+z=-1\). Substituting, we get \(-z^{51}\).

Continue from here.

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#1) Since the surface area is 100x of the Earth, the volume should be \(100^{3/2}=1000\) times of the Earth.

Perfect for the introduction of the 3rd part of the problem.

What happens if you are in the mth dimension?

(:

We still have 1!*2!*3! and so forth, but this time we have triangular numbers, so it would start off like 1!*3!*6!*10!........ and so forth.

Yup! You should be able to do this pretty quickly now (:

First Answer is correct (:

Second answer is just like the first, except you are not adding 1, but it goes from

1, to 3 to 6, to 10....

That should give a pretty good hint (:

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I bashed numbers ending in 1 for mod(143) and got 195 as an answer. Not the elegant solution I hoped for :l

(i) Becuase congruent triangles must have corresponding verticies for them to be labeled as congruent.

(ii) \(\triangle ABC = \triangle ADC\), so it immediately follows that because the triangles are congruent, they have the same angles.

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Algebra, which means "The union of parts" is a broad category of mathematics.

It's different from arithmitic because algebra generally contains an unknown.

For example, in the equation \(2x + 3 = 5\), the unknown is the commonly used variable x. (x is 1.)

Here is a list of categories of mathematics that has 'Algebra' in this. This is from Wikipedia, which has a much more extensive write-up on Algebra than anything you'll see here.

Elementary algebra, the part of algebra that is usually taught in elementary courses of mathematics.

Abstract algebra, in which algebraic structures such as groups, rings and fields are axiomatically defined and investigated.

Linear algebra, in which the specific properties of linear equations, vector spaces and matrices are studied.

Boolean algebra, a branch of algebra abstracting the computation with the truth values false and true.

Commutative algebra, the study of commutative rings.

Computer algebra, the implementation of algebraic methods as algorithms and computer programs.

Homological algebra, the study of algebraic structures that are fundamental to study topological spaces.

Universal algebra, in which properties common to all algebraic structures are studied.

Algebraic number theory, in which the properties of numbers are studied from an algebraic point of view.

Algebraic geometry, a branch of geometry, in its primitive form specifying curves and surfaces as solutions of polynomial equations.

Algebraic combinatorics, in which algebraic methods are used to study combinatorial questions.

Relational algebra: a set of finitary relations that is closed under certain operators.

Just google Algebra. Not that hard.

Let's say, that we want as many weeks to go by before person A hugs person B. Becuase the 8 people sit in a circle, we have to account for rotational symmetry. So, I'll break this up into different cases.

Case 1) A and B sit opposite to eachother.

We can pick any spot for A to sit in, and there is only 1 spot for B to sit in. Since we have 6 seats of any choosing, we have \(\frac{6!}{2} =360\)cases where A and B sit opposite to eachother

*We divide by 2 because there are 2 cases where the order is exactly the same.

The other cases are where B sits 3 and 2 spots away from A. I'll let you work them out.

Let's say that you want to compute n!, also known as n factorial. This expression would be equal to 1 * 2 * 3 * 4 * .......... * (n-1) * n.

For example, if you wanted to compute 4!, then you would have 1*2*3*4 = 24.

0! is regarded as an empty product, so it is 1. 

Another way of representing a factorial is \(\prod_{i=1}^{n} (i)\).

Factorials are often used in complemetary counting, probability, and statistics as such.

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I was unable to get a camera to work, so You'll have to read carfully.

1.Construct a trapezoid with the description of the problem.

2.Name the verticies starting from the top left side, going clockwise, A,B,C and D respectively.

3.Draw altitudes going down from A and B. Name the points where they intersect DC F and E, F is on the left.

4.Let side \(\overline{AB}\) be a, and \(\overline{DC}\) be b. \(\overline{FE}\) is a, and \(\overline{EC}\) = \(\overline{DF}\) = \(\frac{b-a}{2}\).

5.Let the intersection of diagonals be O.

6.\(\triangle DOC\) is a 45-45-90 triangle.

See if you can work it out from there.

In the first equation, multiply by \(-\sqrt[16]{5}+1\) to cancel out all of the nasty roots and you get \(\sqrt[16]{5}x + x = 1\). Simplifying further results in \(x = \sqrt[16]{5}-1\) .\(\)

Since we want \((x+1)^{48}\), we have \((\sqrt[16]{5})^{48}\), which equals \(5^3 = \boxed{125}\)

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Before reading this answer, you might want to take a look at 

It says, that the exterior angle of a polygon with m sides is equal to a regular polygon with g sides. So,

\(\frac{360}{m} = \frac{180(g + 2)}{g}\)

\(360g = 180gm + 360m\)

\(2g = gm + 2m\)

Factoring out g:

\(2g - gm = 2m\)

\(g(2-m) = 2m\)

So therefore, \(g = \frac{2m}{2-m}\).

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"OK, so like the question that i need is "The altitude "a" (in feet) of a plane "t" minutes after liftoff is given by a=3400t+600. How many minutes afterliftoff is the plane at an altitude of 21,000 feet?" Sorry, i didn't put it right earlier."

Maybe you should have put that in the question.

We want to know the number of minutes and we have the value of a, 21,000. Plug in values:

21,000 = 3,400t + 600

20,400 = 3,400t

t = 6

So, it would take \(\boxed{6}\) minutes to achieve an altitude of 21,000 feet.

Since \(\triangle ABC\) is a 30-60-90 triangle and \(\overline{AB}\) is 1, \(\overline{AC}\) is \(\sqrt{3}\) and \(\overline{BC}\) is 2. Let the intersection of the ray \(\overline{AG}\) through \(\overline{BC}\) be D. Becuase \(\overline{AD}\) is a median, it divides \(\overline{BC}\) into 2 equal parts. Therefore, \(\overline{BD}\) is 1. Now, we can say that \(\triangle BAD\) is equilateral because it is isoceles, and one of the angles is 60. As such, \(\overline{AD}\) is 1. 

The centroid divides medians into 2 parts ratio 1:2, so \(\overline{AG}\) is \(\boxed{\frac{2}{3}}\).

First post, How'd I do?

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