To simplify the expression \( Y^{-1/2} \cdot Y^{5/8} \), you can use the property of exponents that states:
\[ a^m \cdot a^n = a^{m+n} \]
In this case, you add the exponents:
\[ Y^{-1/2} \cdot Y^{5/8} = Y^{-1/2 + 5/8} \]
To add \(-1/2\) and \(5/8\), you need a common denominator. The denominator of \(-1/2\) can be converted to 8 by multiplying by 4:
\[ -1/2 = -4/8 \]
Now you can add the fractions:
\[ -4/8 + 5/8 = (5 - 4)/8 = 1/8 \]
So, the exponent becomes:
\[ Y^{1/8} \]
Therefore, the simplified expression is:
\[ Y^{1/8} \]