Question

Apply the properties of integer exponents to simplify the algebraic expression (2ry^3)^4 x (r^2y)^3

Answer 1

To simplify the expression \((2ry^3)^4 \times (r^2y)^3\), we will apply the properties of integer exponents step by step.

1. **Applying the Power of a Product Rule**: This rule states that \((ab)^n = a^n b^n\).
\[
(2ry^3)^4 = (2^4)(r^4)(y^{3 \cdot 4}) = 16r^4y^{12}
\]

2. **Now for the second part**:
\[
(r^2y)^3 = (r^2)^3(y)^3 = r^{2 \cdot 3}(y^3) = r^6y^3
\]

3. **Now we will multiply the two results**:
\[
16r^4y^{12} \times r^6y^3
\]

4. **Combine the coefficients, and apply the product of powers rule for \(r\) and \(y\)**:
\[
16 \times (r^4 \times r^6) \times (y^{12} \times y^3) = 16r^{4+6}y^{12+3} = 16r^{10}y^{15}
\]

Thus, the simplified expression is:
\[
\boxed{16r^{10}y^{15}}
\]