Question

simplify.
$(xz^{3})(-3x^{4}u^{2}z)^{-3}$
write your answer using only positive exponents.

Explanation:

Step1: Apply the negative exponent rule

Recall that \((ab)^n = a^n b^n\) and \(a^{-n}=\frac{1}{a^n}\). So we expand \((-3x^{4}u^{2}z)^{-3}\) as \((-3)^{-3}(x^{4})^{-3}(u^{2})^{-3}(z)^{-3}\).
Now the expression becomes \((xz^{3})\times(-3)^{-3}(x^{4})^{-3}(u^{2})^{-3}(z)^{-3}\).

Step2: Simplify the exponents

Using the power - of - a - power rule \((a^{m})^{n}=a^{mn}\), we have:
\((x^{4})^{-3}=x^{4\times(-3)} = x^{-12}\), \((u^{2})^{-3}=u^{2\times(-3)}=u^{-6}\), \((z)^{-3}=z^{-3}\)
And \((-3)^{-3}=\frac{1}{(-3)^{3}}=\frac{1}{-27}=-\frac{1}{27}\)
So the expression is \(xz^{3}\times(-\frac{1}{27})x^{-12}u^{-6}z^{-3}\)

Step3: Combine like terms

Using the product rule of exponents \(a^{m}\times a^{n}=a^{m + n}\) for the \(x\) and \(z\) terms:
For the \(x\) terms: \(x\times x^{-12}=x^{1+( - 12)}=x^{-11}\)
For the \(z\) terms: \(z^{3}\times z^{-3}=z^{3+( - 3)}=z^{0}=1\) (since \(a^{0}=1,a
eq0\))
The expression now is \(-\frac{1}{27}x^{-11}u^{-6}\times1=-\frac{1}{27}x^{-11}u^{-6}\)

Step4: Convert negative exponents to positive

Using \(a^{-n}=\frac{1}{a^{n}}\), we get:
\(x^{-11}=\frac{1}{x^{11}}\) and \(u^{-6}=\frac{1}{u^{6}}\)
So the simplified expression is \(-\frac{1}{27x^{11}u^{6}}\)

Answer:

\(\boldsymbol{-\frac{1}{27x^{11}u^{6}}}\)