Thực hiện phép tính
a) $\sqrt[]{21-12\sqrt[]{3}}$ – $\sqrt[]{3}$
b) ($\sqrt[]{6}$ +$\sqrt[]{2}$)( $\sqrt[]{3}$ -2)$\sqrt[]{\sqrt[]{3}+2}$
Thực hiện phép tính
a) $\sqrt[]{21-12\sqrt[]{3}}$ – $\sqrt[]{3}$
b) ($\sqrt[]{6}$ +$\sqrt[]{2}$)( $\sqrt[]{3}$ -2)$\sqrt[]{\sqrt[]{3}+2}$
Đáp án:
Giải thích các bước giải:
`a,sqrt{(21-12sqrt{3})}-sqrt{3}`
`=sqrt{9-2.3.2sqrt{3}+12}-sqrt{3}`
`=sqrt{(2sqrt{3}-3)^2}-sqrt{3}`
`=2sqrt{3}-3-sqrt{3}`
`=sqrt{3}-3`
`b, ( sqrt{6} + sqrt{2} )( sqrt{3} -2)(sqrt{(sqrt{3} + 2)})`
`=sqrt{2}(sqrt{3}+1)(sqrt{3}-2)(sqrt{(sqrt{3}+2)})`
`=(sqrt{3}+1)(sqrt{3}-2)(sqrt{(4-2sqrt{3})}`
`=(sqrt{3}+1)(sqrt{3}-2)(sqrt{3}+1)`
`=(4+2sqrt{3})(sqrt{3}-2)`
`=4sqrt{3}-8+6-4sqrt{3}`
`=-2`
Đáp án:
b. -2
Giải thích các bước giải:
\(\begin{array}{l}
b.\sqrt 2 \left( {\sqrt 3 + 1} \right)\left( {\sqrt 3 – 2} \right).\sqrt {2 + \sqrt 3 } \\
= \left( {\sqrt 3 + 1} \right)\left( {\sqrt 3 – 2} \right)\sqrt {4 + 2\sqrt 3 } \\
= \left( {\sqrt 3 + 1} \right)\left( {\sqrt 3 – 2} \right)\sqrt {3 + 2\sqrt 3 .1 + 1} \\
= \left( {\sqrt 3 + 1} \right)\left( {\sqrt 3 – 2} \right)\sqrt {{{\left( {\sqrt 3 + 1} \right)}^2}} \\
= {\left( {\sqrt 3 + 1} \right)^2}\left( {\sqrt 3 – 2} \right)\\
= \left( {3 + 2\sqrt 3 + 1} \right)\left( {\sqrt 3 – 2} \right)\\
= \left( {4 + 2\sqrt 3 } \right)\left( {\sqrt 3 – 2} \right)\\
= 2\left( {\sqrt 3 + 2} \right)\left( {\sqrt 3 – 2} \right)\\
= 2\left( {3 – 4} \right) = – 2
\end{array}\)
\(\begin{array}{l}
a.\sqrt {21 – 12\sqrt 3 } – \sqrt 3 \\
= \sqrt {12 – 2.3.2\sqrt 3 + 9} – \sqrt 3 \\
= \sqrt {{{\left( {2\sqrt 3 – 3} \right)}^2}} – \sqrt 3 \\
= 2\sqrt 3 – 3 – \sqrt 3 \\
= \sqrt 3 – 3
\end{array}\)