Question

Question #bdf2d

Answer 1

`e^1.6=4.9530324244~~4.9530`

Explanation:

You can use a normal calculator, or draw the graph of `y=e^x` on a piece of paper.

graph{e^x [-0.427, 4.573, 2.74, 5.24]}

As you can see here, at `x=1.6`, `y` is around `4.95` or `4.96`. But, you will have to use a calculator to be more accurate.

Answer 2

Alternate way...

`e^(8/5) approx 230771 / 46875 approx 4.9231`

This is just a pure approximate, but actually isnt too close to the actually answer, see other answer for calculator value...

Explanation:

We can use the well-known fact that...

`e^x = 1/(0!) + x/(1!) + x^2/(2!) + x^3/(3!) + ...`

`e^x = sum_(r=0) ^(oo) x^r / (r!)`

We can use the first few terms of `e^x` to approximate `e^1.6`

We know `1.6 = 16/10 = 8/5`

`e^(8/5) approx 1 + (8/5) + (8/5)^2 / 2 + (8/5)^3 / 6 + (8/5)^4 / 24 + (8/5)^5 / 120`

`=> e^(8/5) approx 1 + 8/5 + 32/25 + 256/375 + 512/1875 + 4096/46875`

Simplifying to yield, by finding the common denominator...

`e^(8/5) approx 46875/46875 + 75000/46875 + 60000/46875 + 32000/46875 + 12800 /46875 + 4096/ 46875`

`=> color(orange)(e^(8/5) approx 230771 / 46875 approx 4.9231`

This is an approximate value...