If a= 5^99 x 2^98 and b= 5^72 x 4^38 show ab has 172 digits when written as a ordinary number This is a proof question. Please show step by step.

Step-by-step explanation:

ab

= (5⁹⁹ * 2⁹⁸) * (5⁷² * 4³⁸)

= 2⁹⁸ * 2⁷⁶ * 5⁹⁹ * 5⁷²

= 2¹⁷⁴ * 5¹⁷¹

= 2³ * (2¹⁷¹ * 5¹⁷¹).

The 2¹⁷¹ * 5¹⁷¹ will give 171 ending zeroes as 2 * 5 = 10.

And there is a 8 infront as 2³ = 8.

Hence altogether there are 171 + 1 = 172 digits.

x = central angle apc = measure of minor arc ac

the two angles x and 30 add to 180, so x+30 = 180 and x = 180-30 = 150. the measure of minor arc ac is 150 degrees.

what we need to do from here is find the circumference and then multiply it by x/360 since there are 360 degrees in a circle. in this case, x = 150, so x/360 = 150/360 = 15/36 = 5/12. the shortest distance along the curve from a to c is 5/12 of the distance around the entire circle (aka circumference)

c = circumference

c = pi*d

c = pi*24

c = 24pi

now multiply this by 5/12 to get

distance along minor arc ac = (5/12)*(circumference)

distance along minor arc ac = (5/12)*(24pi)

distance along minor arc ac = 10pi

the formula you can use is

l = (x/360)*pi*d

with

l = length of minor arc

x = central angle

d = diameter

which essentially is the steps we took though a bit differently.

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