Question

simplify. assume all variables are positive.\\(\frac{y^{\frac{6}{5}}}{y^{\frac{11}{8}}}\\)\
write your answer in the form \\(a\\) or \\(\frac{a}{b}\\), where \\(a\\) and \\(b\\) are constants or variable expressions that have no variables in common. all exponents in your answer should be positive.

Explanation:

Step1: Apply exponent rule for division

When dividing exponents with the same base, subtract the exponents: \( \frac{y^a}{y^b} = y^{a - b} \). Here, \( a=\frac{6}{5} \) and \( b = \frac{11}{8} \). So we calculate \( \frac{6}{5}-\frac{11}{8} \).

Step2: Find a common denominator

The common denominator of 5 and 8 is 40. Convert the fractions: \( \frac{6}{5}=\frac{6\times8}{5\times8}=\frac{48}{40} \) and \( \frac{11}{8}=\frac{11\times5}{8\times5}=\frac{55}{40} \).

Step3: Subtract the fractions

\( \frac{48}{40}-\frac{55}{40}=\frac{48 - 55}{40}=\frac{-7}{40} \). So the expression becomes \( y^{-\frac{7}{40}} \).

Step4: Apply negative exponent rule

A negative exponent means taking the reciprocal: \( y^{-n}=\frac{1}{y^n} \). So \( y^{-\frac{7}{40}}=\frac{1}{y^{\frac{7}{40}}} \).

Answer:

\( \frac{1}{y^{\frac{7}{40}}} \)