Please help me with these math problems?

Question

I'm having a tough time with three different problems. Thanks for your help!

1. In the equation (1/5)^m x (1/4)^8 = 1/(2(10)^35), what is m? According to the book, m=35. Could you please explain how they got that answer?

2. Two circles with a diameter of 1 are connected by 15 ft. conveyer belt. What is the distance between the centers of the two circles. The answer is supposed to be (15-pi)/2. Please explain.

3. Finally, I'm having problems with addition and subtraction of numbers with exponents. For example, 10^8-10^2 / 10^7 - 10^3. The answer is 10, but is there an easy way to solve this question w/o multiplying it all out on a calculator?

Thanks for your help!

Answer 1

1)
(1/5)^m x (1/4)^8 = 1/(2(10)^35)
(2/10)^m x (2^(-2))^8 = 2^(-1) x 10^(-35)
2^m x 10^(-m) x 2^(-16) = 2^(-1) x 10^(-35)
2^(m) x 10^(-m) = 2^15 x 10^(-35)
(1/5)^m = 2^15 x 10^(-35)
m = log_1/5(2^15 x 10^(-35)) = 43.6
m does not equal 35, but perhaps there was a typo in the problem.

2) If the circles are a distance x apart from each other, the length of the conveyor belt include 2x to be tangent to the circles on the top and bottom, as well as pi*d/2, where d is diameter, for each semicircle the belt must wrap around. Since d = 1, the total length is 2x + 2(pi*1)/2 = 2x + pi. We also know that the length is 15 feet, so we have 15 = 2x + pi. When you solve for x, you get the given result.

3) Addition and subtraction of exponents cannot be simplified (except by computing them), apart from factoring them. For example, you can write 10^8 - 10^2 as (10^2)(10^6 - 1). In this case, though, you can note that 10^2 is many orders of magnitude smaller than 10^8, and 10^3 is many orders of magnitude smaller than 10^7. Thus, (10^8 - 10^2) / (10^7 - 10^3) is approximately equal to (10^8) / (10^7) = 10^1 = 10. The exact answer is 10.001.

Answer 2

2) Solution:

The cirumference of the Circle is 2 * (Pi) * Radius
Since Radius = Diameter/2 = 1/2

Cirumference = 2 * pi * 1/2 = pi

The Belt covers only half of the circle
therefore, belt covered on each circle = pi / 2
and belt covered on both circles = 2 * pi/2 = pi

Let the distance between the centres be x

15 = 2x + belt covered on both circles
15 = 2x + pi
2x = 15 - pi
x = (15 - pi)/2

3) Solution:

(10^8 - 10^2)/(10^7 - 10^3)

(10^2 (10^6 - 1)) / (10^3 (10^4 - 1))

(10^6 - 1) / (10 (10^4 - 1))

99999 / (10 * 999)

99999 / 9990

Multipling both numerator and denominator by 10^-3 or (0.001)

9.9999 / 0.999

Rounding off

10 / 1

Which is 10

Answer 3

1. for m= 35 (1/4)^8 should be (1/4)^18 not 8

2. length of belt = 2 x + 2 (1/2 *periphery of circle)

x = distance between centers

15 = 2x + 2 (1/2 2 pi 1/2)
15= 2x + pi
x = (15-pi)/2

3.
10^8-10^2 / 10^7 - 10^3
= 10^2 ( 10^6 - 1) / 10^3 (10^4 - 1 )
= (10^3+1)(10^3-1)/(10)(10^2+1)(10^2-1) (using a^2-b^2 = (a+b)(a-b)
= 1001*999/10*101*99
= 10.00099

Answer 4

1.
take log of both sides
log((1/5)^m x (1/4)^8 ) = log(1/(2(10)^35))
m log(1/5) + 8 log(1/4) = log(1/2) + 35 log(1/10)
m log(1/5) = log(1/2) + 35 log(1/10) - 16 log(1/2)
= -15 log(1/2) + 35log (1/5) + 35 log (1/2)
= 20 log(1/2) + 35 log(1/5)
= 20log(1/10) + 15 log(1/5)
m = 20 log(1/10)/log(1/5) + 15 = 43.61


2.
Conveyor belt winds half way around the cylinders. Suppose that the distance between cylinders is d then

2d + 2*(pi*1)/2 = 15
d = (15-pi)/2