Question

What is the square root of 405? and explain it

Answer 1

Find the two perfect square roots closest to 405:

`20^2=400`

`21^2=441`

Write an equation using this info, with the points as `("perfect square", "square root of that perfect square")`:

`(400,20), (441,21)`

Make an equation by finding the slope and the y-int:

`(21-20)/(441-400)=1/41`

`y=1/41x+b`

`20=1/41*400+b`

`b=10.24390`

`y=0.024390x+10.24390`

Plug in `405` as `x`:

`y=0.024390*405+10.24390~~20.09`

About `20.09`

Approximately only, not exact.

Answer 2

`sqrt(405)=9sqrt(5)~~20.1246118`

Explanation:

Extracting the obvious factor of `5` from `405`, we get
`405=5 * 81`
Assuming we recognize `81` as a perfect square (`=9^2`), we can then write
`405 = 5 * 9^2`

then
`sqrt(405)=sqrt(5 * 9^2)`
`color(white)("XXX")=sqrt(5) * sqrt(9^2)`
`color(white)("XXX")=sqrt(5) * 9`
`color(white)("XXX")=9sqrt(5)`

If you need to find an approximation without the radical, you would likely want to use a calculator (there is a "paper-and-pencil" method but it is seldom used/taught).