What is the square root of 405? and explain it

2 Answers
Jensen Z
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.

Alan P.
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).

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