Question

Simplify:

(x * y ^ 3 * z ^ 4) ^ 4

А. x * y ^ 3 * z ^ 16

О В. x * y ^ 3 * z ^ 8

C. x ^ 4 * y ^ 12 * z ^ 16

D. x ^ 5 * y ^ 7 * z ^ 8

Answer 1

To simplify the expression \( (x \cdot y^3 \cdot z^4)^4 \), we apply the power of a product rule, which states that \( (a \cdot b \cdot c)^n = a^n \cdot b^n \cdot c^n \).

Applying this to our expression:

\[
(x \cdot y^3 \cdot z^4)^4 = x^4 \cdot (y^3)^4 \cdot (z^4)^4
\]

Now, we simplify each term:

1. For \( x^4 \), it stays as \( x^4 \).
2. For \( (y^3)^4 \), we use the power rule \( (a^m)^n = a^{m \cdot n} \):
\[
(y^3)^4 = y^{3 \cdot 4} = y^{12}
\]
3. For \( (z^4)^4 \):
\[
(z^4)^4 = z^{4 \cdot 4} = z^{16}
\]

Now we can combine all the parts:
\[
(x \cdot y^3 \cdot z^4)^4 = x^4 \cdot y^{12} \cdot z^{16}
\]

Therefore, the simplified expression is:
\[
C. \quad x^4 \cdot y^{12} \cdot z^{16}
\]

So the answer is **C. \( x^4 \cdot y^{12} \cdot z^{16} \)**.