Express the following fraction in simplest form, only using positive exponents.

start fraction, 3, y, to the power 6 , divided by, 4, left parenthesis, y, to the power minus 1 , right parenthesis, cubed, end fraction
4(y
−1
)
3

3y
6

1 answer

To simplify the given fraction \(\frac{3y^6}{4(y^{-1})^3}\), we can follow these steps:

  1. Simplify the denominator: \[ (y^{-1})^3 = y^{-3} \] So the denominator becomes: \[ 4y^{-3} \]

  2. Rewrite the fraction: Now, we have: \[ \frac{3y^6}{4y^{-3}} \]

  3. Use the property of exponents: We know that dividing by \(y^{-3}\) is the same as multiplying by \(y^3\): \[ \frac{3y^6}{4y^{-3}} = \frac{3y^6 \cdot y^3}{4} \] Combine the exponents of \(y\): \[ y^6 \cdot y^3 = y^{6 + 3} = y^9 \]

  4. Combine everything: Now the expression becomes: \[ \frac{3y^9}{4} \]

  5. Express in simplest form: The fraction \(\frac{3y^9}{4}\) is already in simplest form, using only positive exponents.

Thus, the final answer is: \[ \frac{3y^9}{4} \]