Question

which is a like radical to $sqrt3{54}$ after simplifying? $sqrt{128}$ $sqrt3{128}$ $sqrt3{24}$ $sqrt3{162}$

Explanation:

Step1: Simplify $\sqrt[3]{54}$

Factor 54: $54 = 27\times2 = 3^3\times2$. So $\sqrt[3]{54}=\sqrt[3]{3^3\times2}=3\sqrt[3]{2}$.

Step2: Simplify each option

• Option 1: $\sqrt{128}=\sqrt{64\times2}=8\sqrt{2}$ (square root, different index from cube root, not like radical).
• Option 2: $\sqrt[3]{128}=\sqrt[3]{64\times2}=4\sqrt[3]{2}$ (cube root, radicand part $\sqrt[3]{2}$, same as simplified $\sqrt[3]{54}$? Wait, no, wait: Wait, 128 is $64\times2$, 64 is $4^3$, so $\sqrt[3]{128}=4\sqrt[3]{2}$? Wait no, 64 is $4^3$? No, $4^3=64$, yes. Wait but 54 simplified is $3\sqrt[3]{2}$, so $\sqrt[3]{128}=4\sqrt[3]{2}$, radicand is 2, index 3. Wait but let's check option 4: $\sqrt[3]{162}=\sqrt[3]{81\times2}=\sqrt[3]{27\times3\times2}=3\sqrt[3]{6}$? Wait no, 81 is $3^4$, so 162 is $27\times6$, 27 is $3^3$, so $\sqrt[3]{162}=\sqrt[3]{3^3\times6}=3\sqrt[3]{6}$. Wait no, I made a mistake. Wait 54: $54=27\times2=3^3\times2$, so $\sqrt[3]{54}=3\sqrt[3]{2}$. Now $\sqrt[3]{128}$: 128=64×2=4³×2? No, 4³ is 64, yes, so $\sqrt[3]{128}=4\sqrt[3]{2}$? Wait no, 4³ is 64, 64×2=128, so $\sqrt[3]{128}=\sqrt[3]{4^3\times2}=4\sqrt[3]{2}$. Wait but 54 simplified is $3\sqrt[3]{2}$, so $\sqrt[3]{128}=4\sqrt[3]{2}$, same radicand (2) and index (3). Wait but let's check $\sqrt[3]{162}$: 162=81×2=27×6=3³×6, so $\sqrt[3]{162}=3\sqrt[3]{6}$. $\sqrt[3]{24}=2\sqrt[3]{3}$ (24=8×3=2³×3). $\sqrt{128}$ is square root, index 2, different. Wait so $\sqrt[3]{128}=4\sqrt[3]{2}$, which has the same index (3) and same radicand (2) as $\sqrt[3]{54}=3\sqrt[3]{2}$. Wait but wait, no, wait 128: 128=64×2, 64 is 4³, so $\sqrt[3]{128}=4\sqrt[3]{2}$. So the like radical would have the same index and same radicand after simplification. So $\sqrt[3]{54}=3\sqrt[3]{2}$, $\sqrt[3]{128}=4\sqrt[3]{2}$, so they are like radicals (same index, same radicand). Wait but wait, let's re - check:

Wait, 54: prime factorization is $2\times3^3$, so $\sqrt[3]{54}=\sqrt[3]{3^3\times2}=3\sqrt[3]{2}$.

128: prime factorization is $2^7$, so $\sqrt[3]{128}=\sqrt[3]{2^6\times2}=2^2\sqrt[3]{2}=4\sqrt[3]{2}$. So same index (3) and same radicand ($\sqrt[3]{2}$), so they are like radicals.

Wait but the option 4: $\sqrt[3]{162}$: 162=2×3^4=2×3^3×3, so $\sqrt[3]{162}=3\sqrt[3]{6}$, radicand 6, different. $\sqrt[3]{24}=2\sqrt[3]{3}$, radicand 3, different. $\sqrt{128}$ is square root, index 2, different. So the correct one is $\sqrt[3]{128}$? Wait no, wait I think I made a mistake. Wait 54 is $3^3\times2$, so $\sqrt[3]{54}=3\sqrt[3]{2}$. 162 is $3^3\times6$, so $\sqrt[3]{162}=3\sqrt[3]{6}$. 24 is $2^3\times3$, so $\sqrt[3]{24}=2\sqrt[3]{3}$. 128 is $2^7=2^6\times2$, so $\sqrt[3]{128}=4\sqrt[3]{2}$. So yes, $\sqrt[3]{128}$ simplifies to $4\sqrt[3]{2}$, which has the same index (3) and same radicand (2) as $\sqrt[3]{54}$ (which is $3\sqrt[3]{2}$). So they are like radicals.

Wait but let's check again. Like radicals have the same index and the same radicand (the expression inside the radical) after simplification. So $\sqrt[3]{54}=3\sqrt[3]{2}$ (index 3, radicand 2). $\sqrt[3]{128}=4\sqrt[3]{2}$ (index 3, radicand 2). So they are like radicals.

Answer:

$\boldsymbol{\sqrt[3]{128}}$ (the second option in the top row)