Tutor profile: Wenbin Z.
x^x^x^x^.... = 2 x to the power of x to the power of x infinite times equals 2, what is valve of x
if we cover the first x what does the expression looking like? You're absolution right it's the same expression as the original so x^x^x^... is same as x^(x^x^x^....) and we know x^x^x^.. infinite times equals 2 this implies x^2=2 so x equals sqrt(2)
calculate volume generated by rotating equation f(x) = x^2 around y axis for x from 0 to 1
since this is rotating around y axis let's consider a strip at x with width dx and height of y rotating around y axis. This strip will form a cylindrical shell. The volume of the shell is given by surface area of the shell multiply by the thickness dx which gives 2*pi*r*y*dx with r being x. this results in 2*pi*x*x^2 dx. now we need to add all the strips of x from 0 to 1 so this becomes integral of 2*pi*x^3dx from 0 to 1. the integral equals 2*pi*x^4/4. evaluating this from 0 to 1 gives us 2*pi*1/4. the general expression for evaluating volume generated by function f(x) around y axis is given by integral of 2*pi*x*f(x) dx
If 3x-y=12, what is the valve of 8^x/2^y
There are two ways we can approach this question. First, the equation gives us a relationship between x to y, we can rewrite it as y=3x-12. now we now y in terms of x, we can substitute that into the expression. 8^x/2^y = 8^x/2^(3x-12). let's focus on the denominator 2^(3x-12), from law of exponent we can rewrite it as 2^3x/2^12. Again by law of exponent 2^3x is the same as (2^3)^x which is 8^x. now substitute this into the original expression we have 8^x/(8^x/2^12) which is the same as 1/(1/2^12) which equals 2^12. second approach is to recognize 8^x as (2^3)^x which can be written as 2^3x. this means that 8^x/2^y is same as 2^3x/2^y. using law of exponent this is same as 2^(3x-y). now we know that 3x-y=12 so the answer is 2^12
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