How do we find our approximation for #2.9^5#?

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Answer 1 · 2016-09-24T04:15:37

See explanation

2.9=29/10 is a rational number of the form integer/integer..

And so,29#/1010)^5 = 20511149/100000 is rational.

The exact value in decimals is 205,11149.

Successive rounded approximations are

3-significant digits ( sd:): 205

4-sd: 205.1

5-sd: 205.11

6-sd: 205.111

6-sd: 205.1115.

You can try instead the binomial expansion for

#2.9^5=(3-0.1)^5=3^5(1-1/30)^5#

Answer 2 · 2016-09-24T18:29:21

We can also use the approximation of #y=x^5# around #x=3#.

The derivative of #y# is #dy/dx=5x^4#, so the slope of the tangent line around #x=3# is #5(3^4)=405#.

The point it intersects is #(3,3^5)=(3,243)#.

Thus the tangent line is #y-243=405(x-3)=>y=405x-972#.

Thus, an approximation for #2.9^5# would be:

#y=405(2.9)-972=202.5#.

This compares to the actual value of #2.9^5=205.11149#.