What Question
What to the power of 10 equals 149597870700?
$$\small{\text{$
10^{ \log{(149\ 597\ 870\ 700)} } = 10^{11.1749254120}=149\ 597\ 870\ 700.
$
}}$$
The astronomical unit (AU), which Nature News calls “the rough distance from the Earth to the Sun” and Wikipedia refers to as “the average distance between the Earth and the Sun (roughly speaking)”, has been defined as fixed at 149,597,870,700 metres. This standard was adopted by unanimous vote at the International Astronomical Union’s meeting in Beijing in August 2012.
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$$\\\frac{(x+1)}{(2-x) }< \frac{x}{(3+x)}\\\\
$I am going to multiply both sides by positive numbers and see if that will help$\\\\
(2-x)^2(3+x)^2*\frac{(x+1)}{(2-x) }<(2-x)^2(3+x)^2 \frac{x}{(3+x)}\\\\
(2-x)(3+x)^2(x+1)<(2-x)^2(3+x) x\\\\
(2-x)(3+x)^2(x+1)-(2-x)^2(3+x) x<0\\\\
(2-x)(3+x)\;[(3+x)(x+1)-(2-x) x]<0\\\\
-(x-2)(x+3)\;[(x+3)(x+1)-x(-x+2)]<0\\\\
-(x-2)(x+3)\;[x^2+x+3x+3+x^2-2x]<0\\\\
-(x-2)(x+3)\;[2x^2+2x+3]<0\\\\$$
consider
$$\\2x^2+2x+3=0\\\\
\triangle=4-24<0\\\\
$since the discriminant is 0 there are no roots for this.$\\\\
$the axis of symmetry for $y=2x^2+2x+3\;\;is\;\; \frac{-b}{2a}=\frac{-2}{4}=\frac{-1}{2}$$
the roots are x=2 AND x=-3
the polynomial if set to y and graphed will finish in the bottom right corner because of the - out to front.
I can see that the polynomial will be less than 0 when x>2 and when x<-3
I am sorry i probably have not explained this very well, it was a difficult question for this type.
I did not use the graph but I will draw it now to show you.
https://www.desmos.com/calculator/5xfsalfbbg
Feel free to ask questions ![]()
CPhill's way was probably better for this one. :)