Which of the following is NOT a pair of like surds

$(3 \times \sqrt[2]{80})$ , $(4 \times \sqrt[2]{245})$

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$(2 \times \sqrt[2]{63})$ , $(3 \times \sqrt[2]{12})$

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$(7 \times \sqrt[2]{128})$ , $(4 \times \sqrt[2]{242})$

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$(5 \times \sqrt[2]{108})$ , $(6 \times \sqrt[2]{192})$

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Solution

The correct option is B
$(2 \times \sqrt[2]{63})$ , $(3 \times \sqrt[2]{12})$

$5 \sqrt[2]{108}$ = $5 \sqrt[2]{(36) (3)}$ = $30 \sqrt[2]{3}$ and $6 \sqrt[2]{192}$ = $6 \sqrt[2]{(64) (3)}$ = $48 \sqrt[2]{3}$

both can be expressed as same irrational factor $\sqrt[2]{3}$ hence they are like surds

$3 \sqrt[2]{80}$ = $3 \sqrt[2]{(16) (5)}$ = $12 \sqrt[2]{5}$ and $4 \sqrt[2]{245}$ = $6 \sqrt[2]{(49) (5)}$ = $42 \sqrt[2]{5}$

both can be expressed as same irrational factor $\sqrt[2]{5}$ hence they are like surds

$7 \sqrt[2]{128}$ = $7 \sqrt[2]{(64) (2)}$ = $56 \sqrt[2]{2}$ and $4 \sqrt[2]{242}$ = $4 \sqrt[2]{(121) (2)}$ = $44 \sqrt[2]{2}$

both can be expressed as same irrational factor $\sqrt[2]{2}$ hence they are like surds

$2 \sqrt[2]{63}$ = $2 \sqrt[2]{(9) (7)}$ = $6 \sqrt[2]{7}$ and $3 \sqrt[2]{12}$ = $3 \sqrt[2]{(4) (3)}$ = $6 \sqrt[2]{3}$

both can't be expressed as same irrational factor hence they are not a like surds

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2

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\sqrt{27 a^{3} b^{2} c^{4}} \times \sqrt[3]{128 a^{7} b^{9} c^{2}} \times \sqrt[6]{729 a b^{12} c^{2}}

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