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

Question

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