All Answers 358514 Answers

$$\small{\text{
$
\boxed{
\dfrac{4i^6+3i } {1-2i}
-(9+4i) =\ ?
}
$
}}\\\\
\small{\text{
$
i^2 = -1
\qquad i^4 = i^2 \cdot i^2 = (-1)\cdot (-1) = 1
\qquad i^6 = i^2 \cdot i^2 \cdot i^2 = (-1)\cdot (-1) \cdot (-1) = -1
$
}}\\$$

we set

$$\small{\text{$i^6 = (-1)^3 = -1$}}$$

so we have:

$$\small{\text{
$
\dfrac{4i^6+3i } {1-2i}
-(9+4i) \quad | \quad \boxed{i^6=-1}
$
}}\\\\
\small{\text{
$
=\dfrac{-4+3i } {1-2i}
-(9+4i)
$
}}\\\\
\small{\text{
$
=\dfrac{ (-4+3i) \codt(1+2i) } { (1-2i)\codt(1+2i) }
-(9+4i) \quad | \quad \boxed{(a-ib)\cdot (a+ib) = a^2+b^2}
$
}}\\\\
\small{\text{
$
=\dfrac{ (-4+3i) \codt(1+2i) } { 1^2+2^2 }
-(9+4i)
$
}}\\\\
\small{\text{
$
=\dfrac{ (-4+3i) \codt(1+2i) } { 5 }
-(9+4i)
$
}}\\\\
\small{\text{
$
=\dfrac{ (-4+3i) \codt(1+2i) } { 5 }
-5\cdot\dfrac{ (9+4i) }{5}
$
}}\\\\
\small{\text{
$
=\dfrac{ (-4+3i) \codt(1+2i)-5\codt(9+4i) } { 5 }
$
}}\\\\
\small{\text{
$
=\dfrac{ -4-8i+3i+6i^2 -45 -20i } { 5 }
\quad | \quad \boxed{i^2=-1}
$
}}\\\\
\small{\text{
$
=\dfrac{ -4 -8i + 3i +6\cdot (-1) -45 -20i } { 5 }
$
}}\\\\
\small{\text{
$
=\dfrac{ -4 -8i + 3i -6 -45 -20i } { 5 }
\quad $ we put together
}}\\\\
\small{\text{
$
=\dfrac{ -55 -25i } { 5 }
$
}}\\\\
\small{\text{
$
=-11 -5i
$
}}$$

Thanks anon, you have probably answered this question the way it was intended but this is what was actually asked.

$$\\.06=(188/188+x)(100)\\\\
0.0006=1+x\\\\
0.0006-1=x$$

$${\mathtt{0.000\: \!6}}{\mathtt{\,-\,}}{\mathtt{1}} = {\mathtt{\,-\,}}{\frac{{\mathtt{4\,997}}}{{\mathtt{5\,000}}}} = -{\mathtt{0.999\: \!4}}$$

That is assuming that the boxes fit exactly into the carton with no packing or spaces in between.

here is it....first of all whats the formula for finding SI (simple interest)?

it is $$\frac{p*r*t}{100}$$ 

so in this we will place everything int heir place and solve!

now SI= Amount - Principal 

so SI = 688 - 500

therefore SI =188

now lets place everything

Lets take rate to be 'r'

$$188=\frac{500*r*4}{100}$$

now divide or cut 500 by 100 and multiply by 4

so that gives us

$$188=20r$$

now divide 20 by 188 to get r

$$\frac{188}{20}=r$$

divde!

$$\frac{47}{5}=r$$

$$9.5=r$$

so the rate is 9.5%

to check the answer you can put 9.5 in rate's place and get 188...its correct!

any doubt?

1.

If  y=e^6x 

Find  d(e^6x)/d(x)

$$\\y=e^{6x }\\\\
\frac{dy}{dx}=6e^{6x }$$

-------------------------------------------

$$\\\frac{d}{d(x)}(sin(6x))\\\\
=6cos(6x)\\\\\\
$let me see if I can do this formally using the chain rule$\\\\
y=sin(6x)\\\\
let\;\; u=6x \qquad so \qquad \frac{du}{dx}=6\\\\
y=sin(u)\qquad so \qquad \frac{dy}{du}=cos(u)\\\\
\frac{dy}{dx}=\frac{dy}{du}\times \frac{du}{dx}\\\\
\frac{dy}{dx}=cos(u)\times 6\\\\
\frac{dy}{dx}=cos(6x)\times 6\\\\
\frac{dy}{dx}=6cos(6x)\\\\$$

yeah all that would be really disturbing......i too dont like streile rooms i like a bit messed or filled room.....that one is definitely not my type!

but id like the same room to go on a vacation near a river or a lake when i can see fishes and breathe fresh air!

Thank You!!

Remember that i²= -1

i⁶=(i²)³ ; do this first ,combine like terms and then multply by the conjugate of the denominator, (1+2i).

Should be simple after that, keep combining like terms. I got -11-3i.

Thanks anon, there are some good explanations there. 

cosh hyperbolic cosine

sinh hyperbolic sine

tanh hyperbolic tangent

Good ones to research.

The perimeter is the sum of the sides of a polygon.

(8)x= (15)(12)

$${\mathtt{8}}{\mathtt{\,\times\,}}{\mathtt{x}} = {\mathtt{180}}$$

$${\mathtt{x}} = {\frac{{\mathtt{180}}}{{\mathtt{8}}}}$$

x=22.5

3 3/4 = 15/4

3 3/7 = 24/7

And the product is

15/4 * 24/7 =

15/7 * 24/4 =

15/7 * 6  =  90/7

And the quotient of  8/9 and 1/3   =  (8/9 )  / (1/3)   = (8/9) * (3/1)  = 8/3

And subtracting this from 90/7, we have

90/7  - 8/3 =

(270 - 56) / 21  =

214 / 21

The median is the number in the middle. Since there are an even number of numbers, average the two middle numbers for a median of 15.

Thanks anon.

I am having problems getting my head around this so I'd like another mathematician to take a look please.

acot(sec(acsc -2sqrt 3/2))

I also will interpret this as   acot(sec(acsc( -2sqrt 3/2)))   but you really did need brackets here.

$$acot(sec(acsc( \frac{-2\sqrt 3}{2})))\\\\
=acot(sec(acsc( -\sqrt 3))\\\\
=acot(sec(asin( \frac{1}{-\sqrt 3}))\\\\
NOTE \;\;asin( \frac{1}{-\sqrt 3})\quad $is an angle in the 4th quadrant$\\\\
$And sec of an angle in the 4th quad is positive$\\\\
=acot(\frac{\sqrt3}{\sqrt2})\\\\
=atan(\frac{\sqrt2}{\sqrt3})\\\\
=atan\sqrt{\frac{2}{3}}\\\\$$

$$\underset{\,\,\,\,^{\textcolor[rgb]{0.66,0.66,0.66}{360^\circ}}}{{tan}}^{\!\!\mathtt{-1}}{\left({\sqrt{{\frac{{\mathtt{2}}}{{\mathtt{3}}}}}}\right)} = {\mathtt{39.231\: \!520\: \!483\: \!592^{\circ}}}$$

That is what I get but I'd really like someone else to look at it please.

Even if it is correct, is there an easier way of working it through?

ADDED

I ran this question through Wolfram|Alpha and got the same answer

The terms are already ordered from low to high.

And when we have an even number......we take the middle two values, add them and divide by 2.  So we have...

(14 + 16) / 2   =  30/2 = 15  .....and that's the median

The x coordinate of the midpoint is given by adding the x values and dividing by two.

So.....(5 + 2) / 2  = 7/2

And the same procedure is used to find the y coordinate midpoint.

So.....( 4 + 7) / 2  = 11/2      ....so the midpoint is (7/2, 11/2)

To find the distance between the points we subtract the x coordinates (in any order) and then square the result.

So we have  (5 -2) = 3  ......3^2  = 9

The.....we do the same with the y coordinates - again the order of subtraction is irrelevant.......so we have

(7 - 4)  = 3   .....3^2 = 9

Add these two results and take the square root....so

9 + 9 = 18

√18  = √(9 * 2) =  3√2   .... and that's the distance

Notice that

2 - 5 + 8 - 11 + 14  - 17  +.........+92 - 95 + 98 -101 =

 -3     +   -3     +    -3  + ...........   +    -3      +    -3

And the number of terms is = (101 + 1)/ 3 = 34

So .....there are 17 pairs of terms and each pair sums to -3, we have 17 * -3  = -51 = A