apply the properties of integer exponents to generate an equivalent algebraic expressions to 3r^3y^-4/12r^7y^4

Answer 1

To simplify the expression \(\frac{3r^3y^{-4}}{12r^7y^4}\) using the properties of integer exponents, follow these steps:

1. Simplify the numerical coefficients: \[ \frac{3}{12} = \frac{1}{4} \]

2. Simplify the \(r\) terms using the property \(a^m/a^n = a^{m-n}\): \[ \frac{r^3}{r^7} = r^{3-7} = r^{-4} \]

3. Simplify the \(y\) terms using the same exponent rule: \[ \frac{y^{-4}}{y^{4}} = y^{-4-4} = y^{-8} \]

4. Combine everything: Putting it all together, we have: \[ \frac{3r^3y^{-4}}{12r^7y^4} = \frac{1}{4} r^{-4} y^{-8} \]

Thus, the simplified expression is: \[ \frac{1}{4} r^{-4} y^{-8} \]

Alternatively, you can express the negative exponents as positive by writing them in the denominator: \[ \frac{1}{4} \frac{1}{r^4} \frac{1}{y^8} = \frac{1}{4r^4y^8} \]

So, the final equivalent algebraic expression is: \[ \frac{1}{4r^4y^8} \]