Question

How do you evaluate `96^2*(1/3^2)^3*6^2`?

Answer 1

`4096/9`

Explanation:

Given,

`96^2*(1/3^2)^3*6^2`

Break down the first base into prime numbers.

`=(2^5*3)^2 * (1/3^2)^3 * 6^2`

Simplify.

`=2^10 * 3^2 * 1/3^6 * 6^2`

`=2^10 * 3^2/3^6 * 6^2`

`=2^10 * 3^2/(3^2 * 3^4) * 6^2`

`=2^10 * color(red)cancelcolor(black)(3^2)/(color(red)cancelcolor(black)(3^2) * 3^4) * 6^2`

Break down the last base into prime numbers.

`=2^10 * 1/3^4 * 6^2`

`=2^10 * 1/3^4 * (2*3)^2`

`=2^10 * 1/(3^2*3^2) * 2^2 * 3^2`

`=2^10 * 1/(3^2*color(red)cancelcolor(black)(3^2)) * 2^2 * color(red)cancelcolor(black)(3^2)`

`=2^10 * 1/3^2 * 2^2`

`=1024 * 1/9 *4`

`=4096/9`

Answer 2

The answer can be left in index form. This has more meaning than the actual numbers. `2^12/3^2`

Explanation:

A quicker method would be to work with all the bases at the same time. Change any base to prime factors.

`=(2^5*3)^2 * (1/3^2)^3 * (2*3)^2`

Simplify by removing the brackets.

`=2^10 * 3^2 * 1/3^6 * 2^2*3^2`

Combine like bases by adding the indices:

`=2^12 * 3^4/3^6`

Finally subtract the indices of like bases

`=2^12 /3^2`