To simplify the given expression [tex]\( \frac{6^{\frac{1}{3}}}{6^{\frac{10}{3}}} \)[/tex], we can use the properties of exponents, specifically the rule:
[tex]\[ \frac{a^m}{a^n} = a^{m - n} \][/tex]
In this case, the base [tex]\( a \)[/tex] is 6, [tex]\( m \)[/tex] is [tex]\( \frac{1}{3} \)[/tex], and [tex]\( n \)[/tex] is [tex]\( \frac{10}{3} \)[/tex]. Applying the rule:
[tex]\[ \frac{6^{\frac{1}{3}}}{6^{\frac{10}{3}}} = 6^{\frac{1}{3} - \frac{10}{3}} \][/tex]
Next, simplify the exponent:
[tex]\[ \frac{1}{3} - \frac{10}{3} = \frac{1 - 10}{3} = \frac{-9}{3} = -3 \][/tex]
So we have:
[tex]\[ 6^{\frac{1}{3} - \frac{10}{3}} = 6^{-3} \][/tex]
Now, express [tex]\( 6^{-3} \)[/tex] as a positive exponent by using the property [tex]\( a^{-b} = \frac{1}{a^b} \)[/tex]:
[tex]\[ 6^{-3} = \frac{1}{6^3} \][/tex]
Calculate [tex]\( 6^3 \)[/tex]:
[tex]\[ 6^3 = 6 \times 6 \times 6 = 216 \][/tex]
So:
[tex]\[ 6^{-3} = \frac{1}{216} \][/tex]
Therefore, the simplified form of [tex]\( \frac{6^{\frac{1}{3}}}{6^{\frac{10}{3}}} \)[/tex] is:
[tex]\[ \frac{1}{216} \][/tex]
The correct answer is [tex]\( \boxed{\frac{1}{216}} \)[/tex].
