CalculatorUser

Imaginary number rules,

To the power of -

1: i

2: -1

3: -i

4: 1

5: i

and so on.

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Now we have to find how many times each will occur throughout the sequence.

Since 96th term will be 1, this means that each power of i outcome will occur 24 times. So we have 24i-24-24i+24i

97th term will be i

98th term will be -1

99th term will be -i

So add the three above to the expression we had earlier to get : 25i-25-25i+24i

Simplifies to -25+24i

\(\boxed{-25+24i}\)

Since you are choosing 2 out of 15, you use the formula \(\frac{n!}{r!(n-r)!}\), which is the combination formula

N represents number of items you are choosing out of, and r represents the number of items you are choosing.

Substituting in you will get 105 ways.

Whenever a problem is asking you to choose a certain number of items out of a total number of items, then use the formula above.

Um... I'm only in 7th grade so I just going to solve WITHOUT the complex conjugate and Zbar thingy just for the fun of it.

1. Sqaure both sides.

9z^2-4z^2=1+8

9z^2-4z^2=9

5z^2=9

z^2=9/5

\(\boxed{z=\frac{\sqrt{9}}{5}}\)

Ok... This is the shape (green dots)

I will give you a strategy on how to solve this, quite complicated though.

1. Use the distance formula to find the distance between all the points to find each side length of the five-sided polygon.

2. Divide the shape into three triangles.

3. Use Heron's formula to solve for the area of each triangle (using the side lengths you got from step 1)

4. Add up each triangle's area to find the total area of the shape.

This is definitely not the best way to solve this, and might be very tedious. But it still works.

wow, I totally messed up the exponent rules!!!

Thanks, I thought this would be a hard problem.

Are you trying to solve for one variable? If so, please say it.

This is an algebra problem that can be solved by making an equation with 3 variables.

(1) Make Equations

First equation: "It costs $45 per acre to grow corn, $60 to grow wheat and $50 to grow soybeans" = "A farmer has 1,300 acres of land on which he grows corn, wheat, and soybeans."

So the first equation is this  : \(x+y+z=1300\) the three variables representing HOW MANY acres

Second equation : "It costs $45 per acre to grow corn, $60 to grow wheat and $50 to grow soybeans" = "He has allocated $69,200 for the cost of growing his crops"

So the second equation is this : \(45x+60y+50z=69200\)

Third equation : "Because of market demand the farmer will grow twice as many acres of wheat as of corn."

So the third equation is this : \(y=2x\)

(2) Solve

Now we know the two equations, we can solve.

We can substitute the third equation into the first equation to get \(3x+z=1300\)

Now isolate z.

\(z=1300-3x\)

We can also substitute the third equation into the second equation.

\(165x+50z=69200\)

Now substitute the equation \(z=1300-3x\) into the equation above.

After all this, you should get the answer of 

\(\boxed{x=280}\)

\(\boxed{y=560}\)

\(\boxed{z=460}\)

So in the end, we should have 280 acres of corn, 560 acres of wheat, and 460 acres of soybeans.

Since if the turtle reaches the pond the distance away is 0,  we can make the equation \(\frac{1}{5}-\frac{1}{12}x=0\)

Solving we get 2.4 hours, or 144 minutes

So since we are trying to find the value of the expression, does that mean that the Y value is the answer?