Question

What are all the prime factors of 2025? What is the value of `\sqrt 2025`?

Answer 1

Prime factors of `2025 = 5xx5xx3xx3xx3xx3`

`sqrt(2025)=45`

Explanation:

Here is a decomposition tree for 2045
`color(white)("XXxxxX")color(blue)(2025)`
`color(white)("XXxxxxX")darr`
`color(white)("XXxX")"-------------"`
`color(white)("XXx")darrcolor(white)("xxxxxx")darr`
`color(white)("XXX")color(red)5color(white)("xx")xxcolor(white)("xx")405`
`color(white)("XXxxxxxxxxX")darr`
`color(white)("XXxxxxxxX")"-----------"`
`color(white)("XXxxxxxX")darrcolor(white)("xxxx")darr`
`color(white)("XXxxxxxxX")color(red)5color(white)("x")xxcolor(white)("x")81`
`color(white)("XXxxxxxxxxxxxX")darr`
`color(white)("XXxxxxxxxxX")"--------------"`
`color(white)("XXxxxxxxxX")darrcolor(white)("xxxxxx")darr`
`color(white)("XXxxxxxxxxX")9color(white)("xx")xxcolor(white)("xx")9`
`color(white)("XXxxxxxxxX")darrcolor(white)("xxxxxx")darr`
`color(white)("XXXxxxxxx")"------"color(white)("xxx.x")"------"`
`color(white)("XXXxxxxx")darrcolor(white)("x")darrcolor(white)("xxx")darrcolor(white)("x")darr`
`color(white)("XXxxxxXX")color(red)3 xx color(red)3color(white)("xxxx"color(red)3 xxcolor(red)3`

So we have `2025= color(red)5^2xxcolor(red)3^2xxcolor(red)3^2`
`color(white)("XXX")=(color(red)5xxcolor(red)3xxcolor(red)3)^2`
`color(white)("XXX")=45^2`

Therefore
`color(white)("XXX")sqrt(2025)=sqrt((45^2))=45`