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A property of logarithms is that log(x*y) = log(x) + log(y)

Another is that  log(x^-1) = -log(x)

So:  log(2*a*b) = log(2) + log(a) + log(b)

       log(5/a) = log(5) + log(1/a) = log(5) - log(a)    (as 1/a = a^-1)

      log(10*b) = log(10) + log(b)

Therefore: log(2*a*b) + log(5/a) - log(10*b) =  log(2) + log(a) + log(b) + log(5) - log(a) - log(10) - log(b)

                                                                      = log(2) + log(5) - log(10) 

                                                                      = log(10) - log(10)

                                                                      = 0

I see Melody beat me to it (with a slightly different approach)!

$$\\log 2ab + log 5/a - log10b = 0\\\\
LHS\\
=log\left(\frac{2ab\times \frac{5}{a}}{10b}\right)\\\\
=log\left(\frac{10b}{10b}\right)\\
=log\left(1\right)\\
=0\\
=RHS$$

I assume this might be:

Cot x = cot 118°16’ + 450sin27°30’/(220sin50°56’sin118°16’)

$$$$cot x = cot(118.2666666666666667°) + (450sin(27.5°)/(220sin(50..9333333333333333°)*sin(118.2666666666666667°)) =

cot x = about .843

So x =

cot^-1 (.843) = about 49.85°

So we would solve this like

8x + 4 = 28     ( first out here we would take 4 to the other side of the equal sign and also change it from positive to negative)

8x = 28 - 4       ( now we would simply subtract 4 and 28)

8x = 24    ( and now we will take 8 to the other side and divide it by 24 and so this will give us our x )

x = 24 /8          ( and now dividing them both will simply give us our answer)

x = 3       ANSWER

I hope u have understoodthis by now but if not then pls be free to ask!

i hope this helps!

Thanks Alan.  

5.345621 is the answer

Thanks, Melody......I'm sorry I didn't read the question  closely enough.....I'll go back and provide an edit!!!

Thanks, again....(it's still too early for me, I guess)

We need to find two things here, the dot product of the two vectors and the magnitude of each.

The dot product is <4, 0, 1 > (dot) < 1, 1, 3 > = (4 + 0 + 3) = 7

The magnitude of a = √(4² + 0² + 1²) =√(16 + 1) = √17

And the magnitude of c  = √(1² + 1² + 3²) =√(1 + 1 + 9) = √11

And the angle is given by

cos^-1 (a(dot)c / (llall * llcll)  = cos^-1 ((7) / (√17 * √11) = cos^-1 ((7) / (√187) = cos^-1 (0.5118906968889915) = about  59.2°

What have you answered Chris     These are quadratics.

Oh I just worked out what you did.  I sure found the hard way. 

I am not sure whether I should die of embarasment or die laughing.   Maybe a little of both!

I still don't know why you joined them all together though.

Question 1

ok I am going to think about this as I type.

The solutions to the quadratic formula are 

$$\\x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}\\\\
x=\frac{-b}{2a}\pm \frac{\sqrt{b^2-4ac}}{2a}\\\\\\$$

$$Now if the roots are $2\pm \sqrt5$ then\\\\
$\frac{-b}{2a}=2 \rightarrow b=-4a$\\\\and\\\\
$\frac{\sqrt{b^2-4ac}}{2a}=\sqrt5$\\\\
$\frac{\sqrt{b^2-4ac}}{\sqrt{(2a)^2}}=\sqrt5$\\\\
$\frac{b^2-4ac}{4a^2}=5$\\\\
$b^2-4ac=20a^2$\\\\
$16a^2-4ac=20a^2$\\\\
$a^2-4ac=4a^2$\\\\
$-4ac=4a^2$\\\\$$

$$$-c=a$\\
$c=-a$\\
so the coefficients are a, -4a and -a\\
$y=ax^2-4ax-a$\\
$y=a(x^2-4x-1) $ Where a is a constant and $a\ne 0$$$

In reverse order, we have......

3)    (x - (5 + i)) (x  - (5 - i)) = ((x - 5) - i)) ((x -5) + i)) = (x-5)^2 - i^2 = x^2 -10x + 25 - (-1) = x^2 - 10x +26

2) (x - (3 + 7i))(x -  (3 - 7i)) = (((x-3) - 7i)((x-3) + 7i) = (x-3)^2 - 49i^2 = x^2 - 6x + 9  -49i^2 = x^2 - 6x + 9 + 49 = x^2 - 6x + 58

1) ( x  - (2 + √5)) ( x - (2 - √5)) = ((x - 2) - √5) ((x-2) + √5) = (x-2)^2 - 5 = x^2 - 4x + 4 - 5 = x^2 - 4x - 1

There is nothing to "solve"  .....just  to "expand"

(Thanks to Melody for pointing out my earlier error......)

Thanks for answering this anonymous  

1 3/4 + 2/3 =

= 7/4 + 2/3

= (7/4 + 2/3)

= (7/4 + 2/3)12    this line is not correct.   But the rest is really good 

= (21 + 8)/12

= 29/12

= 2 5/12

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

The only thing I would suggest is that you leave the 1 alone.  Like this

$$\\1\frac{3}{4}+\frac{2}{3}\\\\
=1+\frac{9}{12}+\frac{8}{12}\\\\
=1+\frac{17}{12}\\\\
=1+1\frac{5}{12}\\\\
=2\frac{5}{12}\\\\$$

1 3/4 + 2/3 =

= 7/4 + 2/3

= (7/4 + 2/3)

= (7/4 + 2/3)12

= (21 + 8)/12

= 29/12

= 2 5/12

I think... Correct me if I'm wrong...

Remember order of operation

PEDMAS

Parenthesis, exponents, (division and multiplication) (addition and subtraction)

The exponent ^ must be done before the multiplication *

$$\\\pi*3^2\\
=\pi*9\\
=9\pi\\
\approx 28.27$$

$$7 \frac{1}{6} + 4 \frac{2}{3}\\\\
=11 \frac{1}{6} + \frac{2}{3}\\\\
=11 \frac{1}{6} + \frac{4}{6}\\\\
=11 \frac{5}{6} \\\\$$

A tangent is a line that just touches the curve (function) at a particular point.

A function is said to be increasing at a particular x value if the gradient of the tangent at that point is positive.

A function is said to be decreasing at a particular x value if the gradient of the tangent at that point is negative.

for example 

The red curve is the function that I want to refer to 

The blue tangent has a positive gradient and the green tangent has a negative gradient.

All tangents on the curve where x>1 will have a positive gradients so here the function is increasing.

All tangents on the curve where x<1 will have a negative gradients so here the function is decreasing.

This is true Alan but very few people on this forum have heard of complex numbers.   

$${\mathtt{5\,600}}{\mathtt{\,\times\,}}{\mathtt{16}} = {\mathtt{89\,600}}$$

What is a special right triangle?  This means nothing to me.  Sorry.  

Please use the web2 calc on the home page.   

$$\\-3/2^2+8(-3/2)\\
=-3/4+4*-3\\
=-3/4-12\\
=-(3/4+12)\\
=-12\frac{3}{4}$$

$$\frac{1.9891\times 10^{30}}{1.9008\times 10^{27}}=\frac{1.9891\times 10^{30-27}}{1.9008}=\frac{1.9891\times 10^3}{1.9008}=\frac{1989.1}{1.9008}$$

$${\frac{{\mathtt{1\,989.1}}}{{\mathtt{1.900\: \!8}}}} = {\mathtt{1\,046.454\: \!124\: \!579\: \!124\: \!579\: \!1}}$$

You have not asked us a question so we cannot help you.  

$${\frac{{\mathtt{48}}}{{{\mathtt{4}}}^{{\mathtt{2}}}}}{\mathtt{\,\small\textbf+\,}}{\frac{{\mathtt{3}}}{{\mathtt{5}}}} = {\frac{{\mathtt{18}}}{{\mathtt{5}}}} = {\mathtt{3.6}}$$