Choose the correct simplification of the expression [tex]\left(x y z^2\right)^4[/tex].

A. [tex]x^5 y^5 z^6[/tex]
B. [tex]x^5 y^5 z^8[/tex]
C. [tex]x y z^{16}[/tex]
D. [tex]x^4 y^4 z^8[/tex]



Answer :

To simplify the expression [tex]\(\left(x y z^2\right)^4\)[/tex], follow these steps:

1. Start with the given expression:
[tex]\[ \left(x y z^2\right)^4 \][/tex]

2. Apply the power of a product property, which states that [tex]\((abc)^n = a^n b^n c^n\)[/tex]. Here, [tex]\(a = x\)[/tex], [tex]\(b = y\)[/tex], [tex]\(c = z^2\)[/tex], and [tex]\(n = 4\)[/tex]:

[tex]\[ \left(x y z^2\right)^4 = (x)^4 (y)^4 (z^2)^4 \][/tex]

3. Simplify each factor separately:
- [tex]\(x\)[/tex] raised to the 4th power is [tex]\(x^4\)[/tex]:
[tex]\[ (x)^4 = x^4 \][/tex]
- [tex]\(y\)[/tex] raised to the 4th power is [tex]\(y^4\)[/tex]:
[tex]\[ (y)^4 = y^4 \][/tex]
- [tex]\(z^2\)[/tex] raised to the 4th power. Use the power of a power property, which states that [tex]\((z^m)^n = z^{mn}\)[/tex]. Here [tex]\(m = 2\)[/tex] and [tex]\(n = 4\)[/tex]:
[tex]\[ (z^2)^4 = z^{2 \cdot 4} = z^8 \][/tex]

4. Combine the simplified factors:

[tex]\[ x^4 y^4 z^8 \][/tex]

Therefore, the correct simplification of [tex]\(\left(x y z^2\right)^4\)[/tex] is:

[tex]\[ x^4 y^4 z^8 \][/tex]

Hence, the correct answer is [tex]\(x^4 y^4 z^8\)[/tex].