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Thank you!

Solve for a over the real numbers:
abs(5 a - 4) = a

Split the equation into two possible cases:
5 a - 4 = a or 5 a - 4 = -a

Subtract a - 4 from both sides:
4 a = 4 or 5 a - 4 = -a

Divide both sides by 4:
a = 1 or 5 a - 4 = -a

Add a + 4 to both sides:
a = 1 or 6 a = 4

Divide both sides by 6:
a = 1        or          a = 2/3     Their product is simply: 1 x 2/3 =2/3

3.141592654 

0.000000000410207 + or -

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3.1415926544 10207 - This is the accurate answer if you add them together.

3.1415926535 89793- This is the accurate answer if you subtract them from each other.

So, what is your question? If you want to know how long would it take them, working together, to mow the lawn, then it would be:

1/20 + 1/30 =1/12, or 12 minutes working together.

7438968940885 x 68730857893 x 872956509842976=

446 331 068 129 200 898 026 069 770 535 003 387 680

Thanks for offering, but I will decline courteously. Sometimes, solving puzzles without external aid is exponentially more fulfilling than being spoonfed.

$(-1)^{\frac{1}{2}}\Rightarrow\sqrt{-1}$  is a law of fractional exponents. In general terms, $x^{\frac{a}{b}}=\sqrt[b]{x^a}=\left(\sqrt[b]{x}\right)^a$

Why is this the case? Well, I can attempt to explain it to you.

The law of exponents explains how to handle multiplication of exponents with identical bases. 

$x^3*x^2=x^{3+2}=x^5$

Let's try another example but with fractional exponents.

$3^{\frac{1}{2}}*3^{\frac{1}{2}}=3^{\frac{1}{2}+\frac{1}{2}}=3^1=3$

In this example here, $3^{\frac{1}{2}}$ is a number that when multiplied by itself yields 3. That sounds like the definition of the square root, doesn't it? Therefore, $3^{\frac{1}{2}}=\sqrt{3}$. One can also expound upon this.

The pattern continues. $3^{\frac{1}{3}}$ is a number that when multiplied by itself 3 times yields 3. This is the definition of the cubic root. Then, as you can see, I made a generalization. $3^{\frac{1}{x}}$, when multiplied x times, yields 3. 

This is what WolframAlpha says Melody:

How did you get from $\left(-1\right)^{\frac{1}{2}}$to $\sqrt{-1}$?

2^0.5   IS   sqrt(2)

(-1)^1.5 

$(-1)^{1.5}\\ = (-1)^{\frac{3}{2}}\\ \text{I can look at this in 2 different ways so lets see what happens.}\\ = [(-1)^3]^{\frac{1}{2}}\qquad or \qquad [(-1)^\frac{1}{2}]^{3}\\ = [-1]^{\frac{1}{2}}\qquad or \qquad [\sqrt{-1}]^{3}\\ =\sqrt{-1}\qquad or \qquad [i]^{3}\\ =\pm i \qquad \qquad or \qquad -i\\ $

The only common answer is -i so I guess that is the only correct answer.

Maybe another mathematician would like to comment here......

1) The circumcenter is where the perpendicular bisectors of  of the sides of the triangle meet

So......the circumradius, R ,  can be found as

cos (30)  = (1/2)AB / R    ⇒  R  = (1/2)AB / cos (30)  =

5 / [√3 / 2 ] =  10 / √3 ≈    5.77 units

r(3a)=(3a)^3+(3a)+1

I expect there is more to this question. Is r a constant or a variable?

Here is a pic :

The circle has a radius of 1

B is a vertex of the nonagon......and.....  AE, BD , FB, GC, DB, FB  = 1

AB, BC  = 2 =  two edges of the nonagon

And DBF  = 40°

And ED + DF =  (1/9)"S"

"S"  will consist of  a perimeter of  nine "straight" sides, each with a length of 2, as well as  nine   40° arcs that will total to the perimeter of a circle with a radius of 1, i.e., 2pi

So.....the total perimeter is  9 * 2  + 2 * pi  =  2 [ 9 + pi ]  ≈ 46.27 units 

(-1, 0), (1,0), and (3,1)

Let the center be ( x, y)

So the distance from the center to each point will be the same

So...we have

(x - 1) + y^2  =  (x + 1)^2 + y^2

x^2 - 2x + 1  =  x^2 + 2x + 1

-4x  = 0     so the x coordinate of the center  is    x = 0

And we can find y thusly

(0 -1)^2  + y^2  =  (0 - 3)^2 + (y - 1)^2

1 +  y^2   =  9  + y^2 - 2y + 1

2y =  9   ⇒  y = 9/2  =  4.5

So  the center is  (0, 4.5)

And the radius  is   sqrt ( 1 + 4.5^2)  = sqrt (21.25)

nd the equation of the circle is :   x^2  + (y - 4.5)^2  =  21.25

Here's  a graph : https://www.desmos.com/calculator/puz5c5m6yv

(x + 3)²  +  (y - 3)²  =  6          Let's solve this for  y .

(y - 3)²   =   6 - (x + 3)²

 y - 3   =   ±√[ 6 - (x + 3)² ]

 y   =   ±√[ 6 - (x + 3)² ]  +  3           So....using this value for  y....

$\frac{y}{x}\,=\,\frac{\pm\sqrt{6-(x+3)^2}+3}{x}$

We can say

$Y\,=\,\frac{\pm\sqrt{6-(x+3)^2}+3}{x}$      and we want to know the maximum Y value here.

Here's one way to do this

The line perpendicular to the given line will have the equation

y  = -2(x - 5) + 1

And we can find the intersection of these lines thusly:

-2 (x - 5) + 1   =  (1/2)x + 1

-4x + 20  = x + 2

5x  = 18

x =  18/5        and y  = (1/2)(18/5) + 2  = 38/10  (18/5, 19/5) ⇒ (3.6, 3.8)

So.......from (5,1)   we  went back on x  by [ 5 - 3.6 ] =  1.4  and up on y by [3.8 - 1] = 2.8 to get from (5,1) to (3.6, 3.8) 

So.....from the interesection point of the two lines, we need to do the same thing to determine "Q"

So   we have  ( 3.6 - 1.4 ,  3.8 + 2.8)    ⇒   ( 2.2, 6.6) = "Q"

Here's a graph showing this :  https://www.desmos.com/calculator/zctunpdnrj

First let's find the slope between these two points.

slope  $=\,\frac{\text{change in y}}{\text{change in x}}\,=\,\frac{5-(-16)}{4-(-3)}\,=\,\frac{5+16}{4+3}\,=\,\frac{21}{7}$   =   3

Using a slope of  3  and the point  (4, 5) , the equation of the line in point-slope form is

y - 5  =  3(x - 4)          Distribute the  3 .

y - 5  =  3x - 12          Add  5  to both sides.

y  =  3x - 7          

This is the formula you would use to find the amount of loan that you can afford.

PV=P{[1 + R]^N - 1 / [1 + R]^N} / R

PV=450{[1 + 0.0299/12]^60 - 1 / [1 + 0.0299/12]^60} / [0.0299/12]

PV=$450 x                      55.6661234212.........

PV=$25,049.76 - The amount of loan you can afford

$25,100 - $25,049.76 =$50.24 Your down payment on car loan !!.

13[6^2/(5^2-4^2)+9]

=13[36/(25-16)+9]

=13[36/(9)+9]

=13[4+9]

=13*13

=169