How do you simplify #(7\sqrt { 3} + 2\sqrt { 6} ) ^ { 2}#?

1 Answer
May 12, 2018

#(7sqrt3+2sqrt6)^2=color(blue)(171+84sqrt2#

Explanation:

Simplify:

#(7sqrt3+2sqrt6)^2#

Use the formula for the square of a sum:

#(a+b)=a^2+2ab+b^2#,

where:

#a=7sqrt3#, #b=2sqrt6#

#(7sqrt3+2sqrt6)^2=#

#(7sqrt3)^2+(2xx7sqrt3xx2sqrt6)+(2sqrt6)^2#

Apply multiplication distributive property: #(ab)^2=a^2b^2#

Simplify #(7sqrt3)^2# to #7^2(sqrt3)^2#.

#7^2(sqrt3)^2+(2xx7sqrt3xx2sqrt6)+(2sqrt6)^2#

Simplify #7^2# to #49#.

#49(sqrt3)^2+(2xx7sqrt3xx2sqrt6)+(2sqrt6)^2#

Apply rule: #(sqrta)^2=a#

Simplify #(sqrt3)^2# to #3#.

#49xx3+(2xx7sqrt3xx2sqrt6)+(2sqrt6)^2#

Simplify #49xx3# to #147#.

#147+(2xx7sqrt3xx2sqrt6)+(2sqrt6)^2#

Apply multiplication distributive property: #(ab)^2=a^2b^2#

#147+(2xx7sqrt3xx2sqrt6)+2^2(sqrt6)^2#

Simplify #2^2# to #4#.

#147+(2xx7sqrt3xx2sqrt6)+4(sqrt6)^2#

Apply rule: #(sqrta)^2=a#

Simplify #(sqrt6)^6# to #6#.

#147+(2xx7sqrt3xx2sqrt6)+4xx6#

Simplify #4xx6# to #24#.

#147+(2xx7sqrt3xx2sqrt6)+24#

Apply rule: #sqrtasqrtb=sqrt(axxb)#

Simplify #2xx7sqrt3xx2sqrt6# to #2xx7xx2sqrt(3xx6)#.

#147+(2xx7xx2sqrt(3xx6))+24#

Simplify #sqrt(3xx6)# to #sqrt18#.

#147+(2xx7xx2sqrt18)+24#

Prime factorize #sqrt18#.

#147+(2xx7xx2sqrt(2xx3^2))+24#

Apply rule: #sqrt(a^2)=a#

#147+(2xx7xx2xx3sqrt2)+24#

Simplify #2xx7xx2xx3sqrt2# to #84sqrt2#.

#147+84sqrt2+24#

Simplify #147 +24# to #171#

#171+84sqrt2#