Question

which expression is equivalent to \\(\frac{\sqrt4{6}}{\sqrt3{2}}\\)?\\(\frac{\sqrt4{24}}{2}\\)\\(\frac{\sqrt12{177147}}{3}\\)\\(\frac{\sqrt12{27}}{2}\\)\\(\frac{\sqrt12{55296}}{2}\\)

Explanation:

Step1: Convert radicals to exponents

Rewrite \(\sqrt[4]{6}\) as \(6^{\frac{1}{4}}\) and \(\sqrt[3]{2}\) as \(2^{\frac{1}{3}}\). So the expression becomes \(\frac{6^{\frac{1}{4}}}{2^{\frac{1}{3}}}\).

Step2: Find a common denominator for exponents

The least common multiple of 4 and 3 is 12. Rewrite the exponents: \(6^{\frac{1}{4}} = 6^{\frac{3}{12}}=(6^{3})^{\frac{1}{12}} = 216^{\frac{1}{12}}\) and \(2^{\frac{1}{3}}=2^{\frac{4}{12}}=(2^{4})^{\frac{1}{12}} = 16^{\frac{1}{12}}\). Now the expression is \(\frac{216^{\frac{1}{12}}}{16^{\frac{1}{12}}}\).

Step3: Use the property of exponents \(\frac{a^n}{b^n}=(\frac{a}{b})^n\)

This gives \((\frac{216}{16})^{\frac{1}{12}}=\sqrt[12]{\frac{216}{16}}\). Simplify \(\frac{216}{16}=\frac{27}{2}\)? Wait, no, 216÷8=27, 16÷8=2? No, 216×3=648? Wait, no, let's recalculate. Wait, 6³=216, 2⁴=16. Then \(\frac{6^3}{2^4}=\frac{216}{16}=\frac{27\times8}{2\times8}\)? No, 216÷8=27, 16÷8=2? No, 16 is 2⁴, 216 is 6³=2³×3³. So \(\frac{2^3\times3^3}{2^4}=\frac{3^3}{2}=\frac{27}{2}\). Wait, no, that's not right. Wait, \(\frac{6^3}{2^4}=\frac{216}{16}=\frac{27}{2}\)? 216 divided by 8 is 27, 16 divided by 8 is 2? Yes! So \(\sqrt[12]{\frac{27}{2}}=\frac{\sqrt[12]{27}}{\sqrt[12]{2}}\). But we can rationalize the denominator? Wait, no, let's check the options. Wait, maybe I made a mistake. Wait, let's look at the options. The third option is \(\frac{\sqrt[12]{27}}{2}\). Wait, \(\sqrt[12]{2}=\sqrt[12]{2}\), but 2 is \(2^1 = 2^{\frac{12}{12}}\), so \(\sqrt[12]{2}=2^{\frac{1}{12}}\), and \(\frac{\sqrt[12]{27}}{\sqrt[12]{2}}=\frac{\sqrt[12]{27}}{2^{\frac{1}{12}}}\)? No, wait, maybe another approach. Let's convert each option to 12th root.

Option 1: \(\frac{\sqrt[4]{24}}{2}=\frac{24^{\frac{1}{4}}}{2}=\frac{24^{\frac{3}{12}}}{2^{\frac{12}{12}}}=\frac{\sqrt[12]{24^3}}{2^{12}}\)? No, that's not. Option 3: \(\frac{\sqrt[12]{27}}{2}=\frac{\sqrt[12]{27}}{\sqrt[12]{2^{12}}}=\sqrt[12]{\frac{27}{2^{12}}}\)? No, that's not. Wait, maybe my initial step was wrong. Let's start over.

The original expression is \(\frac{\sqrt[4]{6}}{\sqrt[3]{2}}\). Let's express both radicals with index 12 (LCM of 4 and 3). \(\sqrt[4]{6}=\sqrt[12]{6^3}=\sqrt[12]{216}\), \(\sqrt[3]{2}=\sqrt[12]{2^4}=\sqrt[12]{16}\). So the expression is \(\frac{\sqrt[12]{216}}{\sqrt[12]{16}}=\sqrt[12]{\frac{216}{16}}=\sqrt[12]{\frac{27}{2}}\)? No, 216÷8=27, 16÷8=2? Yes, 216/16=27/2? Wait, 27×2=54, no, 27×8=216, 2×8=16. Yes! So \(\frac{216}{16}=\frac{27}{2}\). Then \(\sqrt[12]{\frac{27}{2}}=\frac{\sqrt[12]{27}}{\sqrt[12]{2}}\). But \(\sqrt[12]{2}=2^{\frac{1}{12}}\), and 2 is \(2^1 = 2^{\frac{12}{12}}\), so \(\sqrt[12]{2}=2^{\frac{1}{12}}\), so \(\frac{\sqrt[12]{27}}{\sqrt[12]{2}}=\frac{\sqrt[12]{27}}{2^{\frac{1}{12}}}\)? No, that's not matching. Wait, maybe the option is \(\frac{\sqrt[12]{27}}{2}\). Wait, let's check the fourth option: \(\frac{\sqrt[12]{55296}}{2}\). Let's calculate 55296. 24³=13824, 24⁴=331776, no. Wait, 6^4=1296, 6^5=7776, 6^6=46656, 6^7=279936, no. Wait, 27×2048=55296? 2048 is 2^11. Wait, 27×2048=55296? 27×2000=54000, 27×48=1296, so 54000+1296=55296. Yes! And 2^11×27=55296? Wait, 2^12=4096, 4096×13.5=55296, no. Wait, 6^3=216, 2^4=16, so \(\frac{6^3}{2^4}=\frac{216}{16}=\frac{27}{2}\), then \(\sqrt[12]{\frac{27}{2}}=\frac{\sqrt[12]{27}}{\sqrt[12]{2}}\). But \(\sqrt[12]{2}\) is 2^(1/12), and 2 is 2^1, so to make the denominator 2, we can multiply numerator and denominator by 2^(11/12): \(\frac{\sqrt[12]{27}\times2^{\frac{11}{12}}}{2}\). But \(\sqrt[12]{27}\times2^{\frac{11}{12}}=\sqrt[12]{27\times2^{11}}=\sqrt[12]{27\times2048}=\sqrt[12]{55296}\). Ah! So \(\frac{\sqrt[12]{27\times2^{11}}}{2}=\frac{\sqrt[12]{55296}}{2}\). Wait, but earlier I thought the third option was correct, but that was a mistake. Wait, let's recalculate:

Original expression: \(\frac{\sqrt[4]{6}}{\sqrt[3]{2}}\)

Convert to 12th roots:

\(\sqrt[4]{6}=\sqrt[12]{6^3}=\sqrt[12]{216}\)

\(\sqrt[3]{2}=\sqrt[12]{2^4}=\sqrt[12]{16}\)

So \(\frac{\sqrt[12]{216}}{\sqrt[12]{16}}=\sqrt[12]{\frac{216}{16}}=\sqrt[12]{\frac{27}{2}}\)

Now, \(\sqrt[12]{\frac{27}{2}}=\frac{\sqrt[12]{27}}{\sqrt[12]{2}}\)

Multiply numerator and denominator by \(\sqrt[12]{2^{11}}\) to rationalize the denominator:

Numerator: \(\sqrt[12]{27}\times\sqrt[12]{2^{11}}=\sqrt[12]{27\times2^{11}}=\sqrt[12]{27\times2048}=\sqrt[12]{55296}\)

Denominator: \(\sqrt[12]{2}\times\sqrt[12]{2^{11}}=\sqrt[12]{2^{12}}=2\)

So the expression is \(\frac{\sqrt[12]{55296}}{2}\), which is the fourth option. Wait, but earlier I thought the third option was correct, but that was an error. Let's check the fourth option: \(\frac{\sqrt[12]{55296}}{2}\). Yes, that's correct. Wait, but let's verify with another method. Let's compute the value of the original expression and the options.

Original expression: \(\frac{\sqrt[4]{6}}{\sqrt[3]{2}}\approx\frac{1.565}{1.26}\approx1.242\)

Fourth option: \(\frac{\sqrt[12]{55296}}{2}\). \(\sqrt[12]{55296}\): 55296^(1/12). Let's see, 2^12=4096, 3^12=531441. 55296 is between 2^12 and 3^12. 2.2^12=2.2^2=4.84, 4.84^2=23.4256, 23.4256^2=548.7, 548.7^2≈301,100. No, that's not. Wait, 55296=27×2048=27×2^11. So (27×2^11)^(1/12)=27^(1/12)×2^(11/12)= (3^3)^(1/12)×2^(11/12)=3^(1/4)×2^(11/12). 3^(1/4)≈1.316, 2^(11/12)≈2^(0.9167)≈1.89. Wait, no, that can't be. Wait, no, 27×2^11=27×2048=55296. Then 55296^(1/12)= (27×2^11)^(1/12)=27^(1/12)×2^(11/12)=3^(3/12)×2^(11/12)=3^(1/4)×2^(11/12). 3^(1/4)≈1.316, 2^(11/12)≈2^(0.9167)≈1.89, so 1.316×1.89≈2.49, divided by 2 is≈1.245, which matches the original expression's approximate value. So the correct option is the fourth one: \(\frac{\sqrt[12]{55296}}{2}\).

Answer:

\(\frac{\sqrt[12]{55296}}{2}\) (the fourth option)