Question

How do you expand `(3+m/3)^5`?

Answer 1

See a solution process below:

Explanation:

Solution for this problem is below:

We can use Pascal's triangle to solve this problem.

The triangle values for the exponent 5 are:

`color(red)(1)color(white)(.........)color(red)(5)color(white)(.........)color(red)(10)color(white)(.........)color(red)(10)color(white)(.........)color(red)(5)color(white)(.........)color(red)(1)`

Therefore `(color(blue)(3) + color(green)(m/3))^5` can be multiplied as:

`color(red)(1)(color(green)((m/3))^0 * color(blue)(3)^5) + color(red)(5)(color(green)((m/3))^1 * color(blue)(3)^4) + color(red)(10)(color(green)((m/3))^2 * color(blue)(3)^3) + color(red)(10)(color(green)((m/3))^3 * color(blue)(3)^2) + color(red)(5)(color(green)((m/3))^4 * color(blue)(3)^1) + color(red)(1)(color(green)((m/3))^5 * color(blue)(3)^0) =>`

`color(red)(1)(color(green)(1) * color(blue)(3)^5) + color(red)(5)(color(green)(m) * color(blue)(3)^3) + color(red)(10)(color(green)(m)^2 * color(blue)(3)^1) + color(red)(10)(color(green)(m)^3 * color(blue)(3)^-1) + color(red)(5)(color(green)(m)^4 * color(blue)(3)^-3) + color(red)(1)(color(green)(m)^5 * color(blue)(3)^-5) =>`

`color(red)(1)(color(green)(1) * color(blue)(243)) + color(red)(5)(color(green)(m) * color(blue)(27)) + color(red)(10)(color(green)(m)^2 * color(blue)(3)) + color(red)(10)(color(green)(m)^3 * color(blue)(1/3)) + color(red)(5)(color(green)(m)^4 * color(blue)(1/27)) + color(red)(1)(color(green)(m)^5 * color(blue)(1/243)) =>`

`243 + 135m + 30m^2 + 10/3m^3 + 5/27m^4 + 1/243m^5`

Or, in standard form:

`1/243m^5 + 5/27m^4 + 10/3m^3 + 30m^2 + 135m + 243`