Answer :
Let's find an expression equivalent to [tex]\(\frac{\sqrt{2}}{\sqrt[3]{2}}\)[/tex].
First, recall the properties of exponents and roots:
[tex]\[ \sqrt{2} = 2^{1/2} \][/tex]
[tex]\[ \sqrt[3]{2} = 2^{1/3} \][/tex]
We need to simplify the expression [tex]\(\frac{\sqrt{2}}{\sqrt[3]{2}}\)[/tex], which translates to:
[tex]\[ \frac{2^{1/2}}{2^{1/3}} \][/tex]
Using the property of exponents that states [tex]\(\frac{a^m}{a^n} = a^{m-n}\)[/tex], we can rewrite the expression as:
[tex]\[ 2^{1/2} \div 2^{1/3} = 2^{1/2 - 1/3} \][/tex]
Next, we need to subtract the exponents:
[tex]\[ 1/2 - 1/3 \][/tex]
To subtract these fractions, find a common denominator. The common denominator for 2 and 3 is 6:
[tex]\[ 1/2 = 3/6 \][/tex]
[tex]\[ 1/3 = 2/6 \][/tex]
Now, subtract:
[tex]\[ 1/2 - 1/3 = 3/6 - 2/6 = 1/6 \][/tex]
So, the expression simplifies to:
[tex]\[ 2^{1/6} \][/tex]
This is equivalent to the sixth root of 2, which can be written as:
[tex]\[ \sqrt[6]{2} \][/tex]
Thus, the correct answer is:
[tex]\[ \sqrt[6]{2} \][/tex]
First, recall the properties of exponents and roots:
[tex]\[ \sqrt{2} = 2^{1/2} \][/tex]
[tex]\[ \sqrt[3]{2} = 2^{1/3} \][/tex]
We need to simplify the expression [tex]\(\frac{\sqrt{2}}{\sqrt[3]{2}}\)[/tex], which translates to:
[tex]\[ \frac{2^{1/2}}{2^{1/3}} \][/tex]
Using the property of exponents that states [tex]\(\frac{a^m}{a^n} = a^{m-n}\)[/tex], we can rewrite the expression as:
[tex]\[ 2^{1/2} \div 2^{1/3} = 2^{1/2 - 1/3} \][/tex]
Next, we need to subtract the exponents:
[tex]\[ 1/2 - 1/3 \][/tex]
To subtract these fractions, find a common denominator. The common denominator for 2 and 3 is 6:
[tex]\[ 1/2 = 3/6 \][/tex]
[tex]\[ 1/3 = 2/6 \][/tex]
Now, subtract:
[tex]\[ 1/2 - 1/3 = 3/6 - 2/6 = 1/6 \][/tex]
So, the expression simplifies to:
[tex]\[ 2^{1/6} \][/tex]
This is equivalent to the sixth root of 2, which can be written as:
[tex]\[ \sqrt[6]{2} \][/tex]
Thus, the correct answer is:
[tex]\[ \sqrt[6]{2} \][/tex]
