How can I find the numerical value of (0.997)^5 to five correct decimal places using the binomial theorem?

0.997 = 1 - 0.003

(0.997)^5 = (1 - 0.003)^5

Binomial theorem gives you

(a + b)^5 =

a^5 + 5a^4b + 10a^3b^2 + 10a^2b^3 + 5ab^4 + b^5

(where the coefficient is found with (5!)/[(n!)(5-n)!]

and the "first term" is actually number zero (n=0)

(remember that 0! = 1)

Use a=1

b = -0.03

a^5 + 5a^4b + 10a^3b^2 + 10a^2b^3 + 5ab^4 + b^5

becomes

1 + 5(1)(-0.003) + 10(1)(-0.003)^2 etc.

= 1 - 0.015 + 0.00009 - 0.00000027 + ...

(I would stop here, as the following terms keep getting smaller, and you only need 5 decimals)

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The signs alternate + - + -

because you are using a negative value (for b)

when it is raised to an even power (b^2, b^4), it becomes positive

when it is raised to an odd power (b, b^3, b^5) it remains negative.

To obtain the result correct to 5 decimal places you need only expand as far as terms in x^2 since x=3x10^-3 so x^2=9x10^-6.

(0.997)^5=(1-0.003)^5

=(1-x)^5

=1-5x+10x^2-...

=1-(5*3x10^-3)+(10*9x10^-6)-...

=1-0.015+0.000090-...

=0.98509.

(1-X)^5

With X=0.003

Binominally expand (1-X)^5

Then substitute in X.

e.g:

1 + 5(-X) + (5x4)/2 x (-X)^2 + (5x4x3)/(2x3) x (-X)^3 + (5x4x3x2)/(2x3x4) x (-X)^4 + (5x4x3x2x1)/(2x3x4x5) x (-X)^5

Which simplifies to:

1 - 5X + 10X^2 - 10X^3 + 5X^4 - X^5

Substitute in X=0.003

1 - 5(0.003) + 10(0.003)^2 - 10(0.003)^3 + 5(0.003)^4 - (0.003)^5

(uses calculator)...

= 0.98509

0.997^5 =>

(1 - 0.003)^5

(a - b)^5 =>

a^5 - 5a^4 * b + 10a^3 * b^2 - 10a^2 * b^3 + 5a * b^4 - b^5

1^5 - 5 * 1^4 * 0.003 + 10 * 1^3 * 0.003^2 - 10 * 1^2 * 0.003^3 + 5 * 1 * 0.003^4 - 0.003^5 =>

1 - 5 * 3 * 10^(-3) + 10 * 3^2 * 10^(-6) - 10 * 3^3 * 10^(-9) + 5 * 3^4 * 10^(-12) - 3^5 * 10^(-15) =>

1 - 15 * 0.001 + 90 * 0.000001 - 270 * 0.000000001 + 405 * 0.000000000001 - 243 * 0.000000000000001 =>

1 - 0.015 + 0.000090 - 0.000000270 + 0.000000000405 - 0.000000000000243 =>

1.000 000 000 000 000 - 0.015 000 000 000 000 + 0.000 090 000 000 000 - 0.000 000 270 000 000 + 0.000 000 000 405 000 - 0.000 000 000 000 243

1.000 000 000 000 000

0.000 090 000 000 000

0.000 000 000 405 000

1.000 090 000 405 000

0.015 000 000 000 000

0.000 000 270 000 000

0.000 000 000 000 243

0.015 000 270 000 243

1.000 090 000 405 000 - 0.015 000 270 000 243

1.000 090 000 405 000

0.015 000 270 000 243

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