Simplify the given expression and re-write it using positive exponents only. ( z 4 u 5 ) 3 ( − z − 2 u − 5 ) − 3 / ( − z u 2 z − 6 ) − 1

To simplify the expression and rewrite it using positive exponents only, let's go through it step by step. The expression given is:

\[
\frac{(z^4 u^5)^3 \cdot (-z^{-2} u^{-5})^{-3}}{(-z u^2 z^{-6})^{-1}}
\]

Let's simplify each part:

### Step 1: Simplify \((z^4 u^5)^3\)

Apply the power rule \((a^m)^n = a^{m \cdot n}\):

\[
(z^4 u^5)^3 = z^{4 \cdot 3} u^{5 \cdot 3} = z^{12} u^{15}
\]

### Step 2: Simplify \((-z^{-2} u^{-5})^{-3}\)

Apply the negative exponent rule \((a^{-m})^{-n} = a^{m \cdot n}\):

\[
(-z^{-2} u^{-5})^{-3} = (-1)^{-3} \cdot z^{2 \cdot 3} \cdot u^{5 \cdot 3} = -z^6 u^{15}
\]

(The \((-1)^{-3}\) remains \(-1\) because raising \(-1\) to any odd power is still \(-1\).)

### Step 3: Simplify the product \((z^{12} u^{15}) \cdot (-z^6 u^{15})\)

Multiply the like terms using the product rule \(a^m \cdot a^n = a^{m+n}\):

\[
z^{12} \cdot (-z^6) = -z^{12+6} = -z^{18}
\]

\[
u^{15} \cdot u^{15} = u^{15+15} = u^{30}
\]

So, the result of the multiplication is:

\[
-(z^{18} u^{30})
\]

### Step 4: Simplify the denominator \((-z u^2 z^{-6})^{-1}\)

Simplify inside the parentheses first:

\[
(-z u^2 z^{-6}) = (-1) \cdot z^{1 + (-6)} \cdot u^2 = -z^{-5} u^2
\]

Applying the negative exponent rule again:

\[
(-z^{-5} u^2)^{-1} = -1 \cdot z^{5} \cdot u^{-2} = -z^5 u^{-2}
\]

### Step 5: Divide the numerator by the simplified denominator

Now, divide:

\[
\frac{-(z^{18} u^{30})}{-z^5 u^{-2}}
\]

Since \(\frac{-A}{-B} = \frac{A}{B}\), the negative signs cancel each other out:

\[
\frac{z^{18} u^{30}}{z^5 u^{-2}}
\]

Applying the division rule \(a^m / a^n = a^{m-n}\):

\[
z^{18-5} = z^{13}
\]

\[
u^{30 - (-2)} = u^{30 + 2} = u^{32}
\]

### Final Simplified Expression

\[
z^{13} u^{32}
\]

Thus, the expression simplified and rewritten using positive exponents only is:

\[
\boxed{z^{13} u^{32}}
\]

---

Would you like further details or have any other questions?

Here are 5 related questions that expand on this concept:

1. How do you simplify expressions with multiple negative exponents?
2. What is the power rule for exponents and when is it applied?
3. How do you handle negative bases when raised to odd versus even powers?
4. Can you simplify \((x^{-3} y^4)^2 \cdot (x^2 y^{-1})^{-3}\)?
5. What is the difference between multiplying and dividing exponents?

**Tip:** Always simplify the inner parentheses and resolve any negative exponents early to avoid mistakes later in the calculations!

Simplify the given expression and re-write it using positive exponents only. ( z^4 u^5 )^3 ( −z^−2 u^−5 )^−3 / ( −z u^2 z^−6 )^−1

Learn how to simplify complex algebraic expressions and rewrite them with positive exponents. This step-by-step solution involves applying the power rule, product rule, and negative exponent rule to simplify the expression (z^4 u^5)^3 (−z^−2 u^−5)^−3 / (−z u^2 z^−6)^−1. Suitable for high school students in Grades 9-12.

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