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\((a^3b^5)^4 = a^{12}b^{20}\)

The number of factors is (12+1)(20+1) = 273.

\(\sqrt{\frac{3}{8}} = \frac{\sqrt{3}}{\sqrt{8}} \cdot \frac{\sqrt{2}}{\sqrt{2}} = \boxed{\frac{\sqrt{6}}{4}}\)

\(\frac{\sqrt{2.5^2 - 0.7^2}}{2.7 - 2.5} =\frac{\sqrt{(5.76)}}{0.2} = \frac{2.4}{0.2} = \boxed{12}\)

When using the quadratic formula, you can look in the square root and that is 9801-2c. So you take the 8001 from the problem and subtract it from 9801 and you get 1800. Then you divide 1800 in half getting 900 which is your answer.

This is a repost, I answered this question already.

\(5(x+2) = x^3 - 2x^2 - 8x\)

\(5(x+2) = x(x-4)(x+2)\)

\(5 = x(x-4)\)                    . (notice that x can't be equal to -2)

\(0 = x^2 - 4x - 5\)

\(\boxed{x = 5, -1}\)

can u explain why 1+p/100

 thanks

I think the problem implicitly assumes that x is positive, since you're dealing with a physical distance. In that case, the answer would be C.

Complementary counting! 

Here is a massive hint! 

Any 2-digit number with a 3 is peachy (and all peachy numbers have at least one 3). There are nine 2-digit numbers with ones digit 3, and there are nine additional 2-digit numbers with tens digit 3 (except from 33, which we've already counted, since its ones digits is 3). Therefore, there are . . .

Here are some reference numbers: \(\sqrt{2021} = 44.955\) and \(\sqrt[3]{2021} = 12.643\).

From 1 to 2021, there are 44 perfect squares and 12 perfect cubes.

From -2021 to -1, there are 12 perfect cubes. (squares can't be negative)

0 is a perfect square and a perfect cube, but you only count it once.

Good point, Chris! ....   Wonder which one it is?   Maybe yours because it didn't say ARC AB = ARC AC  ?    ~EP

log₃₆  = 7-13 + 13/2

LOG₃₆ x    = 1/2

36^log₃₆x = 36^1/2

x = 6

Here's a hint: count all the numbers that are not peachy then subtract from all the 2 digit numbers.

Isolate the log on one side to get \(\log_{36}{x} = -\frac{1}{2}.\)

From there, we get \(x^{-\frac{1}{2}} = 36.\)

Hope this helps.

I interpreted  this one a little differently  than EP  did

I saw  AB , AC  and BC  as  being  chord lengths

Thus  we  have  triangle ABC with  sides of 13, 13  and  24  and  the  wheel  is  the  circumcircle of this triangle

The  circumradius   = 

                             13 * 13  * 24

____________________________________________________   =

sqrt [( 13 + 13 + 24) (13 + 13  -24) ( 24 + 13 - 13) (24 + 13 - 13) ] 

                              4056

 __________________________________   = 

         sqrt [ 50  * 2 * 24 * 24  ] 

                             4056

             ________________  =

                sqrt  [ 57600 ]  

                          4056

                      _______  = 

                          240

                             16. 9

So.....the diameter   =    2 * 16.9  =     33.8  in

Here's a visual representation of EP's answer  :

acutally 4/5. u also have to multiple 20 by 3

Could be they are tangent on opposite sids of the line

  then OP = 3+5 = 8

  if they are tangent on the SAME side of the line ( one circle is inside the other)   then OP would be 5 - 3  = 2 units

I will keep that in mind, thank you!

Hi Cubey,

Welcome to the Web2.0 forum!

I know you are very active here and I like your enthusiasm

but

Please do not answer multiple questions on a single post and please try not to give full answers.

Hints and partial answers (for which you know the full answer) are much better.

This is especially true when it is clear the asker is just looking for quick answers.

I am sure you do not want this forum to be a place where people come with the sole intention of cheating.

\(f(x) = \begin{cases} x^2-4 &\quad \text{if } x \ge -4, \\ x + 3 &\quad \text{otherwise}, \end{cases}\)

then for how many values of    \(x\)   is  \(f(f(x)) = 5\)  ?

---------

You need to look at 2 different cases.

first assume f(x) is less than -4

solve 

5=x+3

x=-2

-2 is not less than -4 so no solutions here.

now assume f(x) >= -4

\(5=x^2-4\\ 9=x^2\\ x=\pm3\)

Both of these are bigger than -4 so they both could have answers.  That is, for this problem

f(x) could be +3 or -3

Now you need to look at each of these for each of the scenarios  x<-4 and x>=-4  and see how many answers you get that are valid.

Circumference = 13 + 13 + 24 = 50  units

pi * d = circumferece

pi * d = 50

d = 50 / pi = 15.92 units

Thanks