Question

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

Answer 1

`(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`