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### Question 6: Simplify each expression and eliminate any negative exponents.

(a) \[
\frac{z^3 z^6}{z^5 z^{-2}}
\]
You submitted the answer "Z6", which was marked incorrect. Likely the issue here involves simplifying the exponents using the properties of exponents and ensuring correct capitalization.

#### Solution:
1. Using the rule \( a^m a^n = a^{m+n} \), simplify both the numerator and denominator:
   \[
   z^3 z^6 = z^{9} \quad \text{and} \quad z^5 z^{-2} = z^{3}
   \]
2. Now simplify the fraction:
   \[
   \frac{z^9}{z^3} = z^{9-3} = z^6
   \]
So, the correct answer is \( z^6 \). Ensure the lowercase "z" is used to match the variable as given in the problem.

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(b) \[
(2a^7)^6
\]
You submitted an incorrect answer for this as well. To simplify:

#### Solution:
1. Apply the exponent rule \( (ab)^n = a^n b^n \), to both the constant and the variables inside the parentheses:
   \[
   (2a^7)^6 = 2^6 a^{7 \cdot 6}
   \]
2. Simplify:
   \[
   2^6 = 64 \quad \text{and} \quad 7 \cdot 6 = 42
   \]
   So, the expression becomes:
   \[
   64a^{42}
   \]
The correct answer is \( 64a^{42} \).

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### Question 7: Simplify each expression.

(a) \[
\sqrt[3]{49x^4y^6z}
\]
You submitted an answer but it was marked incorrect.

#### Solution:
1. Use the property \( \sqrt[n]{a^m} = a^{m/n} \) to apply to each term:
   \[
   \sqrt[3]{49x^4y^6z} = \sqrt[3]{49} \cdot \sqrt[3]{x^4} \cdot \sqrt[3]{y^6} \cdot \sqrt[3]{z}
   \]
2. Simplify each part:
   \[
   \sqrt[3]{49} \quad \text{(which remains in its radical form)}, \quad x^{4/3}, \quad y^{6/3} = y^2, \quad z^{1/3}
   \]
   The expression becomes:
   \[
   \sqrt[3]{49} \cdot x^{4/3} \cdot y^2 \cdot z^{1/3}
   \]
   Alternatively, if a simplified answer form is needed, it might look like:
   \[
   x^{4/3} y^2 z^{1/3} \cdot \sqrt[3]{49}
   \]

Let me know if you need further clarification on any steps, or help with a different part of the problem.

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Would you like more detailed steps or further explanation?

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Here are five related questions:
1. How do you handle negative exponents in fractions?
2. What are the rules for simplifying products of exponents?
3. How do you simplify expressions involving radicals?
4. What is the significance of matching the capitalization of variables?
5. How can you apply exponent rules to both constants and variables?

**Tip**: Always double-check the capitalization and the correct form of your variables when entering answers online!

Simplify each expression and eliminate any negative exponents.

Learn how to simplify expressions with exponents, eliminate negative exponents, and simplify radicals. This problem covers exponent rules and their application in algebra. Suitable for students in Grades 9-12.

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