Factoring out 2^2016 from both the numerator and the denominator:

results in 5/3.

Factoring out 2^2016 from both the numerator and the denominator:

results in 5/3.

Look at where the graph crosses the x axis…then check those x values by seeing that the y value is 0 on the table. Students can also prove their answer by substituting the x values into the equation to see if the result is 0.

Q: If you only have 4s and 5s to make other numbers (such as 4 + 4 + 5 = 13 and 5 + 5 + 5 + 5 + 5 = 25), how many numbers from 1 to 1000 have this property ?

A: 994

Which #s cannot be made? There are 6 numbers that cannot be made:1 2 3 6 7 or 11.

That leaves 994 numbers.

All others can be made from 4s and 5s as 8 is 2 4s and 9 is 4 and 5 and 10 is 5 and 5.

After 12 which is 3 x 4, all other numbers can be made.

See above for 13, 14 is 9+5, 15 is 3 5s, 16 is 4 4 s 17 is 12 +5, 18 is 14 + 4, 19 is 15 +4 and so on and so on…

Q: For how many integer values of x is the value of 4000(2/5)^x an integer?

A: 9

This can be done with guess and check either by hand or with a calculator.

If x = 0, then 4000(2/5)^0 = 4000 so 0 works.

If x = 1, then 4000(2/5)^1 = 1600 (4000 x 0.4) so 1 works.

If x = 0, then 4000(2/5)^0 = 4000 so 0 works.

If x = 1, then 4000(2/5)^1 = 1600 (4000 x 0.4) so 1 works.

If x = 2, then 4000(2/5)^2 = 640 (1600 x 0.4) so 2 works.

If x = 3, then 4000(2/5)^3 = 256 (640 x 0.4) so 3 works.

If x = 4, then 4000(2/5)^4 = 102.4 so 4*does not* work.

We need to also try negative integers so here goes:

If x = 3, then 4000(2/5)^3 = 256 (640 x 0.4) so 3 works.

If x = 4, then 4000(2/5)^4 = 102.4 so 4

We need to also try negative integers so here goes:

If x = -1, then 4000(2/5)^-1 = 10000 (4000 x 2.5) so -1 works.

If x = -2, then 4000(2/5)^-2 = 25000 (10000 x 2.5) so -2 works.

If x = -3, then 4000(2/5)^-3= 62500 (25000 x 2.5) so -3 works.

If x = -4, then 4000(2/5)^-4 = 156250 (62500 x 2.5) so -4 works.

If x = -2, then 4000(2/5)^-2 = 25000 (10000 x 2.5) so -2 works.

If x = -3, then 4000(2/5)^-3= 62500 (25000 x 2.5) so -3 works.

If x = -4, then 4000(2/5)^-4 = 156250 (62500 x 2.5) so -4 works.

If x = -5, then 4000(2/5)^-5 = 390625 (156250 x 2.5) so -5 works.

We have been ok all along because we keep getting even numbers (until x = -4) but now…

If x = -6, then 4000(2/5)^-6 = 976562.5 (390625 x 2.5) so -6 does not work.

There are 9 integers that work:

-5, -4, -3, -2, -1, 0, 1, 2, 3

-5, -4, -3, -2, -1, 0, 1, 2, 3