How do we sole 3^(x+1) = 81^(2x)?
First, we may notice that 81 is a power of three specifically 3^4. We rewrite the equation as 3^(x+1)=(3^4)^(2x). Simplifying we get 3^(x+1)=3^(8x) using the rules of exponents. We can now set the powers equal because the numbers are equal and the bases are the same. So, x+1=8x. Subtracting one from each side gives us 7x=1. Dividing by 7 gives us x=1/7. You should check your result by plugging this into the original equation.
Why might you use De'Moivres theorem to find the product of (3+I*radical (3))^7?
Finding the product (3+i*radical 3))^7 would require someone to multiply out the factor times itself 7 times. If we change the number to trigonometric form using tan (theta)=y/x and x^2+y^2=r^2 we get.:Tan(theta) =(radical (3))/3 or theta is 30 degrees and 3^2 + (rad (3))^2=r^2 or r =rad (12) or 2 rad(3). The trig form is 2rad(3)*(cos 30 +i*sin30). We then apply De'Moivres theorem and get (2 rad(3))^7) (cos 7*30 + i * sin(7*30))
What is the second derivative test used for and what steps should you take to use it?
The second derivative test is used to find local extreme. To use this test you should take the following steps: 1) take the first derivative of the function 2) find the critical points for the first derivative ( f'(x)=0 or f'(x) undefined) 3) find the second derivative f"(x) 4) plug the critical points of the first derivative into the second derivative. If the result is positive, the function is concave up at that point and it is a local minimum at that point. If the result is negative the function is concave down at that point and it is a local minimum at that point.