To simplify the expression
[tex]\[ \frac{y^{\frac{1}{3}}}{y^{\frac{1}{2}} y^{-\frac{3}{2}}}, \][/tex]
we will use the properties of exponents.
1. Combine the exponents in the denominator:
When multiplying expressions with the same base, we add their exponents. Thus, the expression in the denominator
[tex]\[ y^{\frac{1}{2}} y^{-\frac{3}{2}} \][/tex]
can be combined as:
[tex]\[ y^{\left(\frac{1}{2} + \left(-\frac{3}{2}\right)\right)}. \][/tex]
2. Simplify the exponent in the denominator:
[tex]\[ \frac{1}{2} + \left(-\frac{3}{2}\right) = \frac{1}{2} - \frac{3}{2} = \frac{1 - 3}{2} = -1. \][/tex]
So, the denominator simplifies to:
[tex]\[ y^{-1}. \][/tex]
3. Combine the numerator and simplified denominator:
The expression now looks like:
[tex]\[ \frac{y^{\frac{1}{3}}}{y^{-1}}. \][/tex]
4. Use the property of exponents for division:
When dividing expressions with the same base, we subtract the exponents. Thus,
[tex]\[ \frac{y^{m}}{y^{n}} = y^{m - n}. \][/tex]
Applying this to our expression,
[tex]\[ \frac{y^{\frac{1}{3}}}{y^{-1}} = y^{\left(\frac{1}{3} - (-1)\right)}. \][/tex]
5. Simplify the exponent further:
[tex]\[ \frac{1}{3} - (-1) = \frac{1}{3} + 1 = \frac{1}{3} + \frac{3}{3} = \frac{4}{3}. \][/tex]
Therefore, the simplified expression is:
[tex]\[ y^{\frac{4}{3}}. \][/tex]
