Question

Apply the properties of integer exponents to simplify the algebraic expression (2xy3)4⋅(x2y)3 .(1 point)
Responses

x10y15
x superscript 10 baseline y superscript 15 baseline

16x25
16 x superscript 25 baseline

32x10y11
32 x superscript 10 baseline y superscript 11 baseline

16x10y15

Answer 1

To simplify the expression \((2xy^3)^4 \cdot (x^2y)^3\) using the properties of integer exponents, we can follow these steps:

1. Apply the power of a product property: \((ab)^n = a^n b^n\).
2. Apply the exponent rules: \(a^m \cdot a^n = a^{m+n}\).

Let's break it down:

1. Simplify \((2xy^3)^4\):
\[
(2xy^3)^4 = 2^4 \cdot x^4 \cdot (y^3)^4 = 16 \cdot x^4 \cdot y^{12}
\]

2. Simplify \((x^2y)^3\):
\[
(x^2y)^3 = (x^2)^3 \cdot y^3 = x^{6} \cdot y^3
\]

3. Now combine both results:
\[
(2xy^3)^4 \cdot (x^2y)^3 = (16 \cdot x^4 \cdot y^{12}) \cdot (x^{6} \cdot y^{3})
\]

4. Combine the \(x\) and \(y\) terms:
- For \(x\):
\[
x^4 \cdot x^6 = x^{4 + 6} = x^{10}
\]
- For \(y\):
\[
y^{12} \cdot y^3 = y^{12 + 3} = y^{15}
\]

5. Combine everything:
\[
16 \cdot x^{10} \cdot y^{15}
\]

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

The correct response is:
**16x10y15**