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(3√ 4a⁴ b⁵ ) + ( a-c ) / ( b+c )²

(3√ 4⁵ 2⁵ ) + ( 4- ( -5) / ( 2+(-5)²

^{(3√ 1024 * 32) + 1 = 33}   

Because \(\dfrac13 + \dfrac1{3^2} + \dfrac1{3^3} + \cdots = \dfrac{\dfrac13}{1 - \dfrac13} = \dfrac12\),

\(\dfrac12 = 0.\overline{1}_3\)

By a similar approach,

\(\dfrac17 = 0.\overline{1}_8\)

and 

\(\dfrac1{b - 1} = 0.\overline{1}_b\)

The second runner was 40m ahead of the 3rd runner because the first runner finished 40m in front of the 2nd runner and 80m in front of the 3rd. To find the distance in between the 2nd runner and 3rd runner we do the following, 80m - 40m = 40m

You can re-write the expression to be ((2a-3b)^2)^2.  First, you can simplify the inside by using the square of binomials formula: (a+b)^2=a^2+2ab+b^2

This gives 4a^2-12ab+9b^2.  Then you can multiply each term, and get a really big number: 16a^4 - 96a^3*b + 216a^2*b^2 - 216ab^3 + 81b^4

For 1x4 rectangles, the yellow ones, there are 10 of them (going horizontally)

For 4x1 rectangles, the blue ones, there are also 10 (going vertically).

For the 2x2 squares, the green ones, there are 16. 

Add the cases up to get your answer. 

I found these cases by handcounting every rectangle, but if you would like an algebra solution for problems like this with bigger numbers, I could explain that as well.

2 x 32

4 x16

8 x 8  m

  are all possible

1 soap bar = 3/4 of a soap bar + a 3/4 pound weight. The 3/4 pound weight replaces the 1/4 missing of the soap bar. Therefore, in order to find the total weight of the soap bar we have to multiply 3/4 x 4 = 12/4  which we then simplify into 3/1. The total weight of one soap bar is 3 pounds.

Hi! Thank you guys for responding.  I solved it!  I did it like this:

The base 7 number can be "abc_7", and a, b, and c are all digits from 1 to 6, except for b which can also be 0.  Then you can expand it to be 49a + 7b + c.  Then when it is reversed, you get cba.  expanding in base 9, cba_9= 81c+ 9 b + a.  The two expressionss are equal, so you can use the equation 49a + 7b + c = 81c + 9b + a.  Then you move everything to one side and get 80c + 2b - 48a = 0, and you can simplify and get 40 c+ b- 24a = 0.  Then we can solve for b and get  b   = 24a - 40c = 8(3a - 5c). This means that b is divisible by 8, but it is also a base 7 digit so it has to be 0.  then you have 3 a = 5c sp a is divisible by 5 and c is divisible by 3.  They cant be 0 because they are both left digits, so they are 5 and 3.  This means that the answer is 503_7.

Hooray!  Thank you again for helping me :)

Subtract 3y from the first equation to get x = -3y - 4

Substitute x into the second equation to get -9y - 12 - 7y = 36

Simplify: -16y - 12 = 36   -->   -16y = 48   -->   y = -3

Solve for x using the first equation: x = 9 - 4 = 5

Your answer is ( 5 , -3 )

Sum = (1, 190, 4845, 38760, 125970, 184756, 125970, 38760, 4845, 190, 1) = 524,288

Factor the top term and bottom to get:

\(\frac{12u^2(u+4v)}{3(u+4v)}\)

Now we can cancel out the term (u+4v) since the problem tells us that this doesn't equal 0 (division by 0 is undefined)

\(\frac{12u^2}{3}\)

Now simplify to get:

\(4u^2\)

Combine like terms to get: 

\(-7ab^2+9a^3b\)

Let the number of bags for both weights be x.

5x+2x=196

7x=196

x=28

First write the equation in slope-intercept form. y=mx+b, m = slope, b = y-intercept

The slope of two points \((x_{1}, y_{1}) (x_{2}, y_{2})\) is \(\frac {y_{2} - y_{1}}{x_{2}-x_{1}}\). So based on our two points (-3, -5) and (6, 1), we get that our slope equals \(\frac{2}{3}\). Plug this into m and our equation looks like this: y=\(\frac{2}{3}\)x+b.

Now to find b, plug in one of the given points into x and y. I chose to plug in (6,1).

1=\(\frac{2}{3}\)*6+b. We find that b = -3.

This is our equation now: y = \(\frac{2}{3}\)x-3. Since the question is asking for the form Ax+By=C, we multiply by 3 to get rid of the fraction: 3y=2x-9.

Rearrange to get 2x-3y=9, where A=2, B=-3, and C=9. 

Add these numbers up to get A+B+C :)

EDIT: I messed up on some basic arithmetic. I fixed my answer now. 

yeah, but cookies are used.

The site doesn't erase your cookies. Only if you send a DoT signal(do not track) it doesn't store anything.

write out the formula for the triangles.

Then observe the PY and PX.

Put in the numbers to get \( \sqrt[3]{4*4^4*2^5}+1\). 

We can write the numbers inside the square root as powers of 2.

\( \sqrt[3]{2^2*2^8*2^5}+1\).

Now because they all have a common base, you can add the exponents. (To see why, take a basic example like\(2^2*2^3=2*2*2*2*2=2^5\). We are multiplying two 2's and three 2's which makes five threes.)

Adding the exponents, we get \( \sqrt[3]{2^{15}}+1\).

Simplifying, we get \(2^5+1\), which you can figure out on your own.

Thank you!

im not sure plz help

Let the number of people be x. Each person shakes with x-1 people (everyone but themselves).

Since there are x people, there are x(x-1)/2 handshakes. We have to divide by 2 because we counted both of the people that are shaking hands. 

From the problem, we know that x(x-1)/2 = 66. Multiply both sides by 2 to get x(x-1) = 132. 

132 = 12*11, so x = 12

was i right???

The sum of her first four test scores is 80*4=320

To get the maximum possible average of all five tests, she must score a 100 on her last test. 

So we get \(\frac{320+100}{5}\), which is equal to 84. 

ok but in the end i got \(\sqrt[3]{16^42^5}-1\)