Question

What's Square root of 5184?

Answer 1

`72`

Explanation:

Given;

`sqrt5184`

`sqrt(72 xx 72)`

`sqrt72²`

`72^(2 xx 1/2)`

`72`

Answer 2

Demonstrating an intelligent guess approach.

Explanation:

Lets take an 'informed' shot in the dark.

The last digit is 4 and we know that `2xx2=4`

so we could have 2 as our last digit of the root. Using ? to represent the the next digit to the left we have `?2` as a potential number.

Consider the `51` from `5184`

`7xx7=49 larr" May work!"`
`8xx8=64 larr" greater than the 51 from "5184" so will fail"`
`color(white)("dddddddddd.d")" so the 7 x 7 may work"->70xx70`

Putting our guess together we have `72`

Check - splitting the 72 into 70+2

`color(white)("d") 70xx72=5040`
`color(white)("dd") 2xx72 = ul(color(white)(5) 144 larr" Add")`
`color(white)("ddddddddd.") 5184 larr" As required"`

Answer 3

`sqrt(5184) = 2^3 * 3^2 = 72`

Explanation:

Given `5184`

First find the prime factorisation:

`5184 = 2 * 2592`

`color(white)(5184) = 2^2 * 1296`

`color(white)(5184) = 2^3 * 648`

`color(white)(5184) = 2^4 * 324`

`color(white)(5184) = 2^5 * 162`

`color(white)(5184) = 2^6 * 81`

`color(white)(5184) = 2^6 * 3 * 27`

`color(white)(5184) = 2^6 * 3^2 * 9`

`color(white)(5184) = 2^6 * 3^3 * 3`

`color(white)(5184) = 2^6 * 3^4`

Note that all of the factors occur an even number of times, so the square root is exact...

`sqrt(5184) = sqrt(2^6 * 3^4) = 2^3 * 3^2 = 72`