Certainly! Let’s work through the problem step-by-step to divide [tex]\( \frac{x^{\frac{3}{4}}}{x^{\frac{1}{6}}} \)[/tex]:
1. Identify the Exponents:
- The exponent in the numerator is [tex]\( \frac{3}{4} \)[/tex].
- The exponent in the denominator is [tex]\( \frac{1}{6} \)[/tex].
2. Apply the Quotient of Powers Property:
- The quotient of powers property states that for the same base [tex]\( x \)[/tex], [tex]\( \frac{x^a}{x^b} = x^{a-b} \)[/tex]. Here, [tex]\( a = \frac{3}{4} \)[/tex] and [tex]\( b = \frac{1}{6} \)[/tex].
- So, [tex]\( \frac{x^{\frac{3}{4}}}{x^{\frac{1}{6}}} = x^{\left(\frac{3}{4} - \frac{1}{6}\right)} \)[/tex].
3. Simplify the Exponent:
- To subtract the fractions [tex]\( \frac{3}{4} \)[/tex] and [tex]\( \frac{1}{6} \)[/tex], we need a common denominator.
- The least common multiple (LCM) of 4 and 6 is 12.
- Convert [tex]\( \frac{3}{4} \)[/tex] to a fraction with a denominator of 12: [tex]\[ \frac{3}{4} = \frac{3 \times 3}{4 \times 3} = \frac{9}{12} \][/tex].
- Convert [tex]\( \frac{1}{6} \)[/tex] to a fraction with a denominator of 12: [tex]\[ \frac{1}{6} = \frac{1 \times 2}{6 \times 2} = \frac{2}{12} \][/tex].
- Now, subtract the two fractions: [tex]\[ \frac{9}{12} - \frac{2}{12} = \frac{9 - 2}{12} = \frac{7}{12} \][/tex].
4. Write the Final Expression:
- The simplified exponent is [tex]\( \frac{7}{12} \)[/tex].
- Therefore, [tex]\( \frac{x^{\frac{3}{4}}}{x^{\frac{1}{6}}} = x^{\frac{7}{12}} \)[/tex].
So, the answer to the question [tex]\( \frac{x^{\frac{3}{4}}}{x^{\frac{1}{6}}} \)[/tex] is [tex]\( \boxed{x^{\frac{7}{12}}} \)[/tex].
