Comparing Radical Expressions: Finding a Value Greater Than \(\sqrt[3]{12}\)

The question asks us to find which of the given radical expressions has a value greater than \(\sqrt[3]{12}\). To compare radical expressions with different roots (or indices), it is helpful to express them with the same root or to convert them into fractional exponents and compare the values by raising them to a common power.

The base value we are comparing against is \(\sqrt[3]{12}\). Let's evaluate this value approximately. We know that \(2^3 = 8\) and \(3^3 = 27\). Since 12 is between 8 and 27, \(\sqrt[3]{12}\) is between 2 and 3. It is closer to 2 than 3.

A more systematic approach for comparison involves converting the radicals to powers with a common denominator in their exponents. For example, \(\sqrt[n]{a} = a^{1/n}\). To compare \(a^{1/n}\) and \(b^{1/m}\), we can raise both to the power of LCM(n, m). This converts the problem to comparing \(a^{LCM(n, m)/n}\) and \(b^{LCM(n, m)/m}\).

Let's compare \(\sqrt[3]{12}\) with each option:

1. Comparing \(\sqrt[3]{12}\) and \(\sqrt[6]{121}\):

   The roots are 3 and 6. The LCM of 3 and 6 is 6. We raise both values to the power of 6:

   • \((\sqrt[3]{12})^6 = (12^{1/3})^6 = 12^{(1/3) \times 6} = 12^2 = 144\)
   • \((\sqrt[6]{121})^6 = (121^{1/6})^6 = 121^{(1/6) \times 6} = 121^1 = 121\)

   Since \(144 > 121\), it means \((\sqrt[3]{12})^6 > (\sqrt[6]{121})^6\). Therefore, \(\sqrt[3]{12} > \sqrt[6]{121}\). Option 1 is not greater than \(\sqrt[3]{12}\).

2. Comparing \(\sqrt[3]{12}\) and \(\sqrt[12]{33214}\):

   The roots are 3 and 12. The LCM of 3 and 12 is 12. We raise both values to the power of 12:

   • \((\sqrt[3]{12})^{12} = (12^{1/3})^{12} = 12^{(1/3) \times 12} = 12^4\)
   • \(12^4 = 12^2 \times 12^2 = 144 \times 144 = 20736\)
   • \((\sqrt[12]{33214})^{12} = (33214^{1/12})^{12} = 33214^{(1/12) \times 12} = 33214^1 = 33214\)

   Since \(20736 < 33214\), it means \((\sqrt[3]{12})^{12} < (\sqrt[12]{33214})^{12}\). Therefore, \(\sqrt[3]{12} < \sqrt[12]{33214}\). Option 2 is greater than \(\sqrt[3]{12}\).

3. Comparing \(\sqrt[3]{12}\) and \(\sqrt[5]{60}\):

   The roots are 3 and 5. The LCM of 3 and 5 is 15. We raise both values to the power of 15:

   • \((\sqrt[3]{12})^{15} = (12^{1/3})^{15} = 12^{(1/3) \times 15} = 12^5\)
   • \(12^5 = 12^4 \times 12 = 20736 \times 12 = 248832\)
   • \((\sqrt[5]{60})^{15} = (60^{1/5})^{15} = 60^{(1/5) \times 15} = 60^3\)
   • \(60^3 = 60 \times 60 \times 60 = 3600 \times 60 = 216000\)

   Since \(248832 > 216000\), it means \((\sqrt[3]{12})^{15} > (\sqrt[5]{60})^{15}\). Therefore, \(\sqrt[3]{12} > \sqrt[5]{60}\). Option 3 is not greater than \(\sqrt[3]{12}\).

4. Comparing \(\sqrt[3]{12}\) and \(\sqrt[9]{1500}\):

   The roots are 3 and 9. The LCM of 3 and 9 is 9. We raise both values to the power of 9:

   • \((\sqrt[3]{12})^9 = (12^{1/3})^9 = 12^{(1/3) \times 9} = 12^3\)
   • \(12^3 = 12 \times 12 \times 12 = 144 \times 12 = 1728\)
   • \((\sqrt[9]{1500})^9 = (1500^{1/9})^9 = 1500^{(1/9) \times 9} = 1500^1 = 1500\)

   Since \(1728 > 1500\), it means \((\sqrt[3]{12})^9 > (\sqrt[9]{1500})^9\). Therefore, \(\sqrt[3]{12} > \sqrt[9]{1500}\). Option 4 is not greater than \(\sqrt[3]{12}\).

Based on the comparisons, only Option 2, \(\sqrt[12]{33214}\), is greater than \(\sqrt[3]{12}\).

Summary of Comparisons

Conclusion

Comparing \(\sqrt[3]{12}\) with each option by raising them to the least common multiple of their roots shows that \(\sqrt[12]{33214}\) is the only value among the options that is greater than \(\sqrt[3]{12}\).

Revision Table: Understanding Radical Expressions

Additional Information: Comparing Different Roots

Comparing radical expressions like \(\sqrt[n]{a}\) and \(\sqrt[m]{b}\) is a common task. The most reliable method involves rewriting them so they have the same index. This is done by finding the least common multiple (LCM) of the indices n and m. Let k = LCM(n, m).

Rewrite the expressions using the property \(a^{1/n} = a^{(k/n) \times (1/k)} = (a^{k/n})^{1/k}\):

• \(\sqrt[n]{a} = a^{1/n} = (a^{k/n})^{1/k} = \sqrt[k]{a^{k/n}}\)
• \(\sqrt[m]{b} = b^{1/m} = (b^{k/m})^{1/k} = \sqrt[k]{b^{k/m}}\)

Now, compare the values by comparing the radicands under the common root k: compare \(a^{k/n}\) and \(b^{k/m}\). The expression with the larger radicand under the same root k is the greater value.

For example, to compare \(\sqrt[3]{12}\) and \(\sqrt[5]{60}\):

• Indices are 3 and 5. LCM(3, 5) = 15.
• Rewrite \(\sqrt[3]{12}\) with index 15: \(\sqrt[3]{12} = \sqrt[15]{12^{15/3}} = \sqrt[15]{12^5} = \sqrt[15]{248832}\).
• Rewrite \(\sqrt[5]{60}\) with index 15: \(\sqrt[5]{60} = \sqrt[15]{60^{15/5}} = \sqrt[15]{60^3} = \sqrt[15]{216000}\).
• Compare the radicands: 248832 and 216000.
• Since \(248832 > 216000\), \(\sqrt[15]{248832} > \sqrt[15]{216000}\), which means \(\sqrt[3]{12} > \sqrt[5]{60}\).

This method is equivalent to raising both original expressions to the power of the LCM of their indices, as demonstrated in the step-by-step solution above.