How do you simplify #sqrt3 - sqrt27 + 5sqrt12 #?
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#sqrt(3) - sqrt(27) + 5sqrt(12)#
#sqrt(3) - sqrt(9*3)+5sqrt(12)# #color(blue)(" 27 factors into "9*3)#
#sqrt(3)- 3sqrt(3)+5sqrt(12)# #color(blue)(" 9 is a perfect square, so take a 3 out")#
#sqrt(3)-3sqrt(3)+5sqrt(4*3)# #color(blue)(" 12 factors into " 4*3)#
#sqrt(3)-3sqrt(3)+5*2sqrt(3)# #color(blue)(" 4 is a perfect square, so take a 2 out")#
#sqrt(3)-3sqrt(3)+10sqrt(3)# #color(blue)(" To simplify, "5*2=10)#
Now that everything is in like terms of #sqrt(3)#, we can simplify:
#sqrt(3)-3sqrt(3)+10sqrt(3)#
#-2sqrt(3)+10sqrt(3)# #color(blue)(" Subtraction: "1sqrt(3)-3sqrt(3)=-2sqrt(3))#
#8sqrt(3)# #color(blue)(" Addition: "10sqrt(3)+(-2sqrt(3))=8sqrt(3))#
#√3−√27+5√12#
#=√3−3√3+5√12#
#=√3−3√3+10√3#
#=8√3#
- Simplify each surd to create a 'like' surd, when each number under the root sign is the same. This allows us to calculate the addition of the surds.
- We first simplify √27 to 9√3 = √27 and then simplify the number outside the root sign to = 3 (The square root) this gives us 3√3
- Then we simplify 5√12 to the √12 = 2√3 and then multiply this by 5 = 10√3
- Because each surd is now in the 'like' surd form we can carry out simple addition to complete the equation.
- #=√3−3√3+10√3#
#=8√3#
Given: #sqrt(3) - sqrt(27) + 5 sqrt(12)#
Simplify using perfect squares and the rule: #sqrt(m*n) = sqrt(m)*sqrt(n)#
Some perfect squares are:
#2^2 = 4#
#3^2 = 9#
#4^2 = 16#
#5^2 = 25#
#6^2 = 36#
...
#sqrt(3) - sqrt(27) + 5 sqrt(12) #
#= sqrt(3) - sqrt(9*3) + 5 sqrt(4*3)#
#= sqrt(3) - sqrt(9)sqrt(3)+ 5 sqrt(4)sqrt(3)#
#= sqrt(3) - 3sqrt(3) + 5 * 2sqrt(3)#
#= sqrt(3) - 3sqrt(3) + 10sqrt(3)#
Since all terms are alike they can be added or subtracted:
#sqrt(3) - sqrt(27) + 5 sqrt(12) = 8 sqrt(3)#
