What is the square root of 405? and explain it

2 Answers
Mar 7, 2018

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.

Mar 7, 2018

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).