rút gọn biểu thức `\sqrt{19+6\sqrt{2}}-\sqrt{19-6\sqrt{2}}`
2 câu trả lời
Đáp án+Giải thích các bước giải:
`\sqrt{19+6\sqrt{2}}-\sqrt{19-6\sqrt{2}}`
`=\sqrt{18+1+6\sqrt{2}}-\sqrt{18+1-6\sqrt{2}}`
`=\sqrt{(3\sqrt{2})^2+1+2.3\sqrt{2}}-\sqrt{(3\sqrt{2})^2+1-6\sqrt{2}}`
`=\sqrt{(3\sqrt{2}+1)^2}-\sqrt{(3\sqrt{2}-1)^2}`
`=|3\sqrt{2}+1|-|3\sqrt{2}-1|`
`=3\sqrt{2}+1-3\sqrt{2}+1`
`=2`
Đặt : $A=\sqrt[]{19+6\sqrt[]{2}}-\sqrt[]{19-6\sqrt[]{2}}>0$
$=>A^2=19+6\sqrt[]{2}+19-6\sqrt[]{2}-2\sqrt[]{(19+6\sqrt[]{2})(19-6\sqrt[]{2})}$
$=38-2\sqrt[]{19^2-(6\sqrt[]{2})^2}$
$=38-2\sqrt[]{361-72}$
$=38-2\sqrt[]{289}$
$=38-2.17$
$=38-34=4$
$=>A =\sqrt[]{4}=2$
Hay $\sqrt[]{19+6\sqrt[]{2}}-\sqrt[]{19-6\sqrt[]{2}}=2$