Verified answer

(a)

(1 + 3x)^8 = 1 + 8(3x) + (8*7/2) (3x)^2 + (8*7*6/6) (3x)^3 + ...

= 1 + 24x + 252x^2 + 1512x^3 + ...

You are correct so far, well done

(b)

(1.003)^8 = (1 + 3x)^8

when

1.003 = 1 + 3x

3x = 0.003

x = 0.001

Now just substitute this into your expansion from (a)

1 + 24x + 252x^2 + 1512x^3 = 1 + 24 [0.001] + 252 [0.001]^2 + 1512 [0.001]^3

= 1 + 0.024 + 0.000252 + 0.000001512

= 1.024253512

Hence

(1.003)^8 = 1.0242535 to 8 significant figures

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(1+3x)^4(1+3)4=(1+3x)(1+3x)(1+3x)(1+3x)(1+3x)(1+3x)(1+3x)(1+3x)