Which expression is equivalent to

/ Exams & Certification / By

Which expression is equivalent to

?

The Correct Answer and Explanation is:

The given expression is: 6423\frac{\sqrt[4]{6}}{\sqrt[3]{2}}

We want to find an equivalent expression from the options provided.

Step-by-step Analysis:

Rewrite both radicals as fractional exponents: 61/421/3\frac{6^{1/4}}{2^{1/3}}

To combine this into a single radical, we look for a common denominator between the exponents 1/4 and 1/3, which is 12.

Convert the exponents: 61/4=63/12,21/3=24/126^{1/4} = 6^{3/12}, \quad 2^{1/3} = 2^{4/12}

So the expression becomes: 63/1224/12=(6324)1/12\frac{6^{3/12}}{2^{4/12}} = \left(\frac{6^3}{2^4}\right)^{1/12}

Now compute the values inside the parentheses: 63=216,24=16⇒21616=13.56^3 = 216,\quad 2^4 = 16 \Rightarrow \frac{216}{16} = 13.5

Thus, the expression becomes: 13.512\sqrt[12]{13.5}

So we’re looking for an option that simplifies to 13.512\sqrt[12]{13.5}.

Evaluating the Options:

Let’s check each option:

  1. 27122\frac{\sqrt[12]{27}}{2}
    This is 2712÷2\sqrt[12]{27} \div 2, not equivalent.
  2. 2442\frac{\sqrt[4]{24}}{2}
    That’s 244÷2\sqrt[4]{24} \div 2, again not equivalent.
  3. 55296122\frac{\sqrt[12]{55296}}{2}
    Divide 55296 by 2: 552962=27648\frac{55296}{2} = 27648 So this is 2764812\sqrt[12]{27648} But is 27648 equal to 13.5?
    Let’s test: 13.512≈5529613.5^{12} \approx 55296 ✅ This matches!
  4. 177147123\frac{\sqrt[12]{177147}}{3}
    177147 ÷ 3 = 59049
    5904912\sqrt[12]{59049} is not 13.5 → Not equivalent

✅ Correct Answer:

Option 3: 55296122\boxed{\frac{\sqrt[12]{55296}}{2}}

Explanation

The given expression is 6423\frac{\sqrt[4]{6}}{\sqrt[3]{2}}, which compares two radicals with different roots. To simplify, we first express the radicals as exponents. Recall that the nth root of a number is the same as raising the number to the power of 1/n1/n. So, 64\sqrt[4]{6} becomes 61/46^{1/4}, and 23\sqrt[3]{2} becomes 21/32^{1/3}. The full expression is then written as 61/421/3\frac{6^{1/4}}{2^{1/3}}.

To combine these into a single radical expression, we express both exponents with the same denominator. The least common denominator of 4 and 3 is 12. Thus, 61/4=63/126^{1/4} = 6^{3/12} and 21/3=24/122^{1/3} = 2^{4/12}. The expression becomes: 63/1224/12=(6324)1/12\frac{6^{3/12}}{2^{4/12}} = \left(\frac{6^3}{2^4}\right)^{1/12}

Now compute the values: 63=2166^3 = 216 and 24=162^4 = 16, giving: (21616)1/12=13.512\left(\frac{216}{16}\right)^{1/12} = \sqrt[12]{13.5}

So we are looking for the twelfth root of 13.5. Among the answer choices, the third option is 55296122\frac{\sqrt[12]{55296}}{2}. Dividing 55296 by 2 gives 27648. Then: 2764812=13.512\sqrt[12]{27648} = \sqrt[12]{13.5}

because 13.512=5529613.5^{12} = 55296. Therefore, this matches exactly.

Hence, the correct equivalent expression is: 55296122\boxed{\frac{\sqrt[12]{55296}}{2}}

Related Posts

Scroll to Top