Rút gọn các biểu thức sau:
`A=\sqrt{4+2\sqrt{3}}+\sqrt{4-2\sqrt{3}}`
`B=\sqrt{6+2\sqrt{5}}+\sqrt{6-2\sqrt{5}}`
`C=\sqrt{7+4\sqrt{3}}-\sqrt{7-4\sqrt{3}}`
Rút gọn các biểu thức sau: `A=\sqrt{4+2\sqrt{3}}+\sqrt{4-2\sqrt{3}}` `B=\sqrt{6+2\sqrt{5}}+\sqrt{6-2\sqrt{5}}` `C=\sqrt{7+4\sqrt{3}}-\sqrt{7-4\sqrt{3}
By Athena
`A=\sqrt{4+2\sqrt3}+\sqrt{4-2\sqrt3}`
`=\sqrt{3+2\sqrt3+1}+\sqrt{3-2\sqrt3+1}`
`=\sqrt{(\sqrt3+1)^2}+\sqrt{(\sqrt3-1)^2}`
`=\sqrt3+1+\sqrt3-1`
`=2\sqrt3`
`B=\sqrt{6+2\sqrt5}+\sqrt{6-2\sqrt5}`
`=\sqrt{5+2\sqrt5+1}+\sqrt{5-2\sqrt5+1}`
`=\sqrt{(\sqrt5+1)^2}+\sqrt{(\sqrt5-1)^2}`
`=\sqrt5+1+\sqrt5-1`
`=2\sqrt5`
`C=\sqrt{7+4\sqrt3}+\sqrt{7-4\sqrt3}`
`=\sqrt{3+2.2\sqrt3+4}+\sqrt{3-2.2\sqrt3+4}`
`=\sqrt{(\sqrt3+2)^2}+\sqrt{(\sqrt3-2)^2}`
`=\sqrt3+2+2-\sqrt3`
`=4`
Đáp án + Giải thích các bước giải:
`A=sqrt(4+2sqrt3)+sqrt(4-2sqrt3)=sqrt(3+2sqrt3+1)+sqrt(3-2sqrt3+1)` $\\$ `= sqrt[(sqrt3)^2 + 2*sqrt3*1+1^2] + sqrt[(sqrt3)^2-2*sqrt3*1+1^2]` $\\$ `= sqrt[(sqrt3+1)^2] + sqrt[(sqrt3-1)^2]` $\\$ `= |sqrt 3 + 1| + |sqrt3 – 1| = sqrt3+1+sqrt3-1=2sqrt3`
`B=sqrt(6+2sqrt5)+sqrt(6-2sqrt5) = sqrt(5+2sqrt5+1)+sqrt(5-2sqrt5+1)` $\\$ `= sqrt[(sqrt5)^2+2*sqrt5*1+1^2] + sqrt[(sqrt5)^2-2*sqrt5*1+1^2]` $\\$ `= sqrt[(sqrt5+1)^2] + sqrt[(sqrt5-1)^2]` $\\$ `= |sqrt5+1|+|sqrt5-1|=sqrt5+1+sqrt5-1=2sqrt5`
`C=sqrt(7+4sqrt3)-sqrt(7-4sqrt3) = sqrt(4+4sqrt3+3) – sqrt(4-4sqrt3+3)` $\\$ `= sqrt(2^2+2*2*sqrt3+(sqrt3)^2) – sqrt(2^2-2*2sqrt3+(sqrt3)^2` $\\$ `= sqrt[(2+sqrt3)^2] – sqrt[(2-sqrt3)^2]` $\\$ `= |2 + sqrt3| – |2 – sqrt3| = 2 + sqrt3 – (2 – sqrt3) =2+sqrt3-2+sqrt3=2sqrt3 `