Halloween Scribe Post - New Unit: Exponents

Today's class, we were divided into groups and started off solving for "x" given 2-cosx = 2sin^2x
I started off with changing the sin^2x into 1-cos^2x, then multiply it by 2. The moved everything onto the left side, getting 2cos^2x - cosx = 0. Then factor cosx out of it which you'll then get cosx(2cosx -1) = 0. From there you can tell that cosx = 0 and cosx = 1/2 which means x = pi/2, 3pi/2 and x = pi/3 and 5pi/3.

We were then introduced to the new unit; only a very light introduction. We were asked to express the numbers 2, 3, 4/9, and 1/4 in the the forms of exponents. For example, 2 can be express as 4^(1/2), 3 can be expressed as 6561^(1/8), 4/9 can be expressed as (16/81)^(1/2) and 1/4 can be expressed as 2^(-2). You can also give numbers negative exponents, which you would then have to turn the base of the number into it's reciprocal. For example; 4^(1/2) can also be expressed as (1/4)^(-1/2). Notice the base of the number that was previously 4, became 1/4 and the exponent of 1/2 became -1/2. We were then asked to express 1/4 in 10 other ways for tomorrow's class.

Afterwards, we got a little deeper into the subject; now having to find the exponents in the given equation 2^x = 64. That one was a no brainer. But then there was 27^(2x-1) = 3. The way you solve that is to rewrite the equation with both sides of the equal sign having the same base number; for example 27 = 3^3, then you have the base number of 3 on both sides. So (3^3)^(2x-1) = 3^1. You then take the exponents alone, (3)(2x-1) = 1 **note that the "3" in that is the exponent not the base, and the "1" is from the right side of the equation because 3 is the same thing as 3^1**. Then you simply solve for "x";;

3(2x-1) = 1
6x - 3 = 1
6x = 4
x = 4/6 or 2/3

So there you have solved for "x" in the equation 27^(2x-1) = 3

We were also given in the set, 3^x = 1/27. Remembering that negative exponents give you recipricols, the answer must be (-3).

Lastly, we were given this set to solve for x:

4^(1-x) = 8 49^(x-2) = 7 √7 25^2x = 5^(x+6)

the answers are as follows:


4^(1-x) = 8 (2^2)^(1-x) = 2^3
(2)(1-x) = 3
2 - 2x = 3
2x = -1
x = =1/2

49^(x-2) = 7 √7
(7^2)^(x-2) = (7^1)(7^(1/2))
(7^2)^(x-2) = 7^(3/2)
(2)(x-2) = (3/2)
2x - 4 = 3/2
2x = 4 + 3/2
x = 11/4

25^2x = 5^(x+6)
(5^2)^(2x) = 5^(x+6)
(2)(2x) = x+6
3x = 6
x = 2

And that was everything that we covered in Halloween's class, that I'm surprised everyone showed up to. Tomorrow's scribe can be Sergio, he can cover All Saints Day's class for everyone.