Question

simplify \\(\left( \frac{2x^3 y z^4}{4x^5 y^3 z} \
ight)^{-2}\\)

Explanation:

Step1: Simplify the fraction inside the parentheses

First, simplify the coefficients and the variables with exponents separately. For the coefficients, we have $\frac{2}{4}=\frac{1}{2}$. For the $x$ terms, using the rule of exponents $a^m\div a^n = a^{m - n}$, we get $x^{3}\div x^{5}=x^{3 - 5}=x^{-2}$. For the $y$ terms, $y\div y^{3}=y^{1 - 3}=y^{-2}$. For the $z$ terms, $z^{4}\div z = z^{4 - 1}=z^{3}$. So the fraction inside the parentheses simplifies to $\frac{1}{2}x^{-2}y^{-2}z^{3}$.

Step2: Apply the outer exponent of -2

Using the rule $(ab)^n=a^n b^n$ and $(a^m)^n=a^{m\times n}$, we apply the exponent -2 to each part. For the coefficient, $(\frac{1}{2})^{-2}=\frac{1^{-2}}{2^{-2}}=\frac{2^{2}}{1^{2}} = 4$ (since $a^{-n}=\frac{1}{a^{n}}$ or $\frac{1}{a^{-n}}=a^{n}$). For the $x$ term, $(x^{-2})^{-2}=x^{(-2)\times(-2)}=x^{4}$. For the $y$ term, $(y^{-2})^{-2}=y^{(-2)\times(-2)}=y^{4}$. For the $z$ term, $(z^{3})^{-2}=z^{3\times(-2)}=z^{-6}$.

Step3: Combine the results

Now, multiply all the parts together: $4\times x^{4}\times y^{4}\times z^{-6}$. Since $z^{-6}=\frac{1}{z^{6}}$, we can write this as $\frac{4x^{4}y^{4}}{z^{6}}$.

Answer:

$\frac{4x^{4}y^{4}}{z^{6}}$