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o k thank you

I hope I could help.

[1 x 2^5 + [0 x 2^4] + [0 x 2^3] +[0 x 2^2] + [0 x 2^1] + [0 x 2^0] = 32

1                  0                0               0               0               0

Then the smallest number in base ten that has 6 digits in base 2:

32 =100000

Can someone please answer this question? Please....thanks!

The smallest prime number is 2. 

We can check if \(3^{12}+5^{13}+7^{14}+11^{15}\equiv0\pmod2\)

Since the product of any amount of odd numbers is odd:

3 is odd

3² is odd

3³ is odd

...

We know that the expression is:

odd + odd + odd + odd = even 

Since the expression is even, it must be divisible by 2. 

Hey oscar.a1551!

The area cross section of ABC is equal to the are of ABC.

Since ABC is a right triangle, we can calculate its area. 

We know that the hypotenuse is 17, and the height is 8, but we need to figure out the base. 

Using the Pythagorean Theorem: a^2 + b^2 = c^2 

In a right triangle, the sum of the square of the two legs is equal to the square of the hypotenuse. 

If the base is x, 8^2 + x^2 = 17^2 , x = 15

The area of the triangle is half of the base times height.

8 * 15 / 2 = 60.

I hope this helped,

Gavin

Hey oscar.a1551!

The area of a cross section that is parallel to face ABCD has the equal area of ABCD.

Assuming ABCD is a rectangle, we can use calculate the area of ABCD.

Given 6 and 12, the area of the rectangle is 6*12 = 72 

Since the cross section has equal area to ABCD, its area is also 72. 

I hope this helped,

Gavin

I am thinking of two numbers in the ratio . The difference between the two numbers is . What is the sum of the two numbers?

Hello guest!

\(\color{BrickRed}if\ (relationship\ R=\frac{a}{b}=a-b)\ then\\ relationship\ R=a-b\\ a=R+b\\ relationship\ R=\frac{a}{b}\\ b=\frac{a}{R}\\ a+b=R+b+\frac{a}{R}\\ \color{blue}a+b=\frac{R^2+a+bR}{R}\)

Did you mean something like that?

Greatings

  !

Let       \(y=(0.\overline{12})+(0.\overline{345})\).

When y is written out as a repeating decimal, what is the sum of the digits in a single repeating period? (In other words, what is the sum of the digits covered by the repeat bar?)

Let t =  0.1212121212........

then 100t= 12.12121212.....

100t - t = 12

99t=12

t= 12/99

0.1212121212... = 12/99

Now do the same thing with 0.345345345.....

only since there are 3 digits repeating you will have to multiply by 1000.

Now if you understand what i have written you can finish it yourself.

(a)

Find the z score

[ 20555 - 20500 ] / 55  =  .1

This z score  equates to  .8413  =  84.13% of the days

(b)

Find the z  score

[20610 - 20500] / 55 =  2

This score equates to .9772.....so the % days that the factory  produced  20610 or more skateboards  =

1 - .9772 = .0228  =  2.28%  of the days

(c)

Find the z score

[20445 - 20500 ]  / 55  = -1

This z score equates to .1587 = 15.87%  of the days

Notice the pattern : 

1, 4, 3, -1, -4, -3, 1, 4, 3, -1, -4, -3

The series has a cycle of 6.....and the sum of these terms  = 0

So

2014 mod 6 =  4

So we will have   2010/6  = 335 of these cycles  and  the last  4 terms after that are   1,4,3  -1

So....the sum  =  7

Center of  P =  (-9, 1)     radius  =  5 

Center  of Q  =( -2, -3)   radius  = 6

So...the answers are

7

right

4

below

a longer radius

y=1/2x^2+6x+24     factor out the 1/2

y  = (1/2) [ x^2 + 12x + 48  ] completet te square on x

y  = (1/2)  [ x^2  + 12 x  + 36  + 48 - 36 ]    factor the first three terms in the parenthses, simplify the rest

y =  (1/2)  [ ( x + 6)^2    + 12 ]      

y  =(1/2) ( x + 6)^2  + 6    

y - 6  =  (1/2)( x + 6)^2    multiply both sides by 2

2 ( y - 6)  = (x+ 6)^2

The vertex  is   ( --6, 6)

To find the directrix

4p  = 2

p = 2/4

p =  1/2

The parabola turns upward...so the  directrix   is   y  = 6  - 1/2   =  5.5

The perimeter of the cross section is the perimeter of rectangle DEKL.

perimeter of DEKL  =  DE + EK + KL + LD

(where  DE ,  EK ,  KL , and  LD  are in meters)

KL  =  12     and     DE  =  12

To find the length of EK, look at right triangle EJK.

By the Pythagorean theorem,

EJ²  +  JK²   =   EK²

                                       Plug in  5  for  EJ  and  4  for  JK .

 5²   +   4²   =    EK²
                                       Simplify the left side of the equation.

25  +  16   =   EK²

41   =   EK²

                                       Take the positive square root of both sides.

√41  =  EK

So.....

EK  =  √41     and     LD  =  √41

perimeter of DEKL  =  DE + EK + KL + LD

                                                                       Plug in the values of  DE ,  EK ,  KL , and  LD .

perimeter of DEKL  =  12 + √41 + 12 + √41

                                                                       Combine like terms.

perimeter of DEKL  =  24 + 2√41

Steps to finding the directrix given the equation of the parabola:
1) Find p-value
2) Find vertex
3) Sketch a diagram and label it (optional, but helpful)
steps 1 and 2 can be done in any order

x=-b/2a is the formula used to find the axis of symmetry of parabolas
Refer to this for more info on the axis of symmetry and what it is: https://mathbitsnotebook.com/Algebra1/Quadratics/QDGraphInfo.html

They're not asking for the number of possible paths the frog can take, they're asking for the number of possible final positions

Hope this helps!

12^3=1728

A)84.1%

B)97.5%

C)2.5%

Welcome back max ;)

Here's a hint for 3: -2 will always be a root of the polynomial

EDIT: whoops, tried to post this as an answer

Hi Max. Sorry, no LaTex!

2.         Solve for x:

x^2 + 7 x - 5 = 5 sqrt(x^3 - 1)

x^2 + 7 x - 5 = 5 sqrt(x^3 - 1) is equivalent to 5 sqrt(x^3 - 1) = x^2 + 7 x - 5:

5 sqrt(x^3 - 1) = x^2 + 7 x - 5

Raise both sides to the power of two:

25 (x^3 - 1) = (x^2 + 7 x - 5)^2

Expand out terms of the left hand side:

25 x^3 - 25 = (x^2 + 7 x - 5)^2

Expand out terms of the right hand side:

25 x^3 - 25 = x^4 + 14 x^3 + 39 x^2 - 70 x + 25

Subtract x^4 + 14 x^3 + 39 x^2 - 70 x + 25 from both sides:

-x^4 + 11 x^3 - 39 x^2 + 70 x - 50 = 0

The left hand side factors into a product with three terms:

-(x^2 - 8 x + 10) (x^2 - 3 x + 5) = 0

Multiply both sides by -1:

(x^2 - 8 x + 10) (x^2 - 3 x + 5) = 0

Split into two equations:

x^2 - 8 x + 10 = 0 or x^2 - 3 x + 5 = 0

Subtract 10 from both sides:

x^2 - 8 x = -10 or x^2 - 3 x + 5 = 0

Add 16 to both sides:

x^2 - 8 x + 16 = 6 or x^2 - 3 x + 5 = 0

Write the left hand side as a square:

(x - 4)^2 = 6 or x^2 - 3 x + 5 = 0

Take the square root of both sides:

x - 4 = sqrt(6) or x - 4 = -sqrt(6) or x^2 - 3 x + 5 = 0

Add 4 to both sides:

x = 4 + sqrt(6) or x - 4 = -sqrt(6) or x^2 - 3 x + 5 = 0

Add 4 to both sides:

x = 4 + sqrt(6) or x = 4 - sqrt(6) or x^2 - 3 x + 5 = 0

Subtract 5 from both sides:

x = 4 + sqrt(6) or x = 4 - sqrt(6) or x^2 - 3 x = -5

Add 9/4 to both sides:

x = 4 + sqrt(6) or x = 4 - sqrt(6) or x^2 - 3 x + 9/4 = -11/4

Write the left hand side as a square:

x = 4 + sqrt(6) or x = 4 - sqrt(6) or (x - 3/2)^2 = -11/4

Take the square root of both sides:

x = 4 + sqrt(6) or x = 4 - sqrt(6) or x - 3/2 = (i sqrt(11))/2 or x - 3/2 = 1/2 (-i) sqrt(11)

Add 3/2 to both sides:

x = 4 + sqrt(6) or x = 4 - sqrt(6) or x = 3/2 + (i sqrt(11))/2 or x - 3/2 = -(i sqrt(11))/2

Add 3/2 to both sides:

x = 4 + sqrt(6) or x = 4 - sqrt(6) or x = 3/2 + (i sqrt(11))/2 or x = 3/2 - (i sqrt(11))/2

[4 + sqrt(6)] * [4 - sqrt(6)]= 10

[Courtesy of Mathematica 11 Home Edition]

whoops, sorry forgot to put that in my answer thx MW