Answer :
Let's simplify the given expression:
[tex]\[ \frac{\sqrt[4]{6}}{\sqrt[5]{6}} \][/tex]
First, we need to express the roots as exponents. Recall that [tex]\( \sqrt[n]{a} = a^{\frac{1}{n}} \)[/tex].
Therefore, we can rewrite the expression as:
[tex]\[ \frac{6^{\frac{1}{4}}}{6^{\frac{1}{5}}} \][/tex]
When dividing powers with the same base, we subtract the exponents:
[tex]\[ 6^{\frac{1}{4}} \div 6^{\frac{1}{5}} = 6^{\left( \frac{1}{4} - \frac{1}{5} \right)} \][/tex]
Next, we need to perform the subtraction in the exponent:
[tex]\[ \frac{1}{4} - \frac{1}{5} \][/tex]
To subtract these fractions, we first need a common denominator. The least common multiple of 4 and 5 is 20. We convert each fraction to have this common denominator:
[tex]\[ \frac{1}{4} = \frac{5}{20} \quad \text{and} \quad \frac{1}{5} = \frac{4}{20} \][/tex]
Now we subtract the fractions:
[tex]\[ \frac{5}{20} - \frac{4}{20} = \frac{1}{20} \][/tex]
Thus, the exponent simplifies to:
[tex]\[ 6^{\left( \frac{1}{20} \right)} \][/tex]
Therefore, the expression simplifies to:
[tex]\[ 6^{\frac{1}{20}} \][/tex]
So, the simplified form of [tex]\(\frac{\sqrt[4]{6}}{\sqrt[5]{6}}\)[/tex] is:
[tex]\[ 6^{\frac{1}{20}} \][/tex]
Hence, the correct answer is:
[tex]\[ \boxed{6^{\frac{1}{20}}} \][/tex]
[tex]\[ \frac{\sqrt[4]{6}}{\sqrt[5]{6}} \][/tex]
First, we need to express the roots as exponents. Recall that [tex]\( \sqrt[n]{a} = a^{\frac{1}{n}} \)[/tex].
Therefore, we can rewrite the expression as:
[tex]\[ \frac{6^{\frac{1}{4}}}{6^{\frac{1}{5}}} \][/tex]
When dividing powers with the same base, we subtract the exponents:
[tex]\[ 6^{\frac{1}{4}} \div 6^{\frac{1}{5}} = 6^{\left( \frac{1}{4} - \frac{1}{5} \right)} \][/tex]
Next, we need to perform the subtraction in the exponent:
[tex]\[ \frac{1}{4} - \frac{1}{5} \][/tex]
To subtract these fractions, we first need a common denominator. The least common multiple of 4 and 5 is 20. We convert each fraction to have this common denominator:
[tex]\[ \frac{1}{4} = \frac{5}{20} \quad \text{and} \quad \frac{1}{5} = \frac{4}{20} \][/tex]
Now we subtract the fractions:
[tex]\[ \frac{5}{20} - \frac{4}{20} = \frac{1}{20} \][/tex]
Thus, the exponent simplifies to:
[tex]\[ 6^{\left( \frac{1}{20} \right)} \][/tex]
Therefore, the expression simplifies to:
[tex]\[ 6^{\frac{1}{20}} \][/tex]
So, the simplified form of [tex]\(\frac{\sqrt[4]{6}}{\sqrt[5]{6}}\)[/tex] is:
[tex]\[ 6^{\frac{1}{20}} \][/tex]
Hence, the correct answer is:
[tex]\[ \boxed{6^{\frac{1}{20}}} \][/tex]
