https://file.aiquickdraw.com/mathbot/file/17263063236247icbmutn.jpg

Let's go over the questions in the image you uploaded.

1. **Simplify \((x^3y^5)(x^6y^7)\)**  
   A) \(x^{15}y^5\)  
   B) \(x^9y^{12}\)  
   C) \(x^3y^{35}\)  
   D) \(x^9y^5\)

2. **Simplify \(\frac{x^4y^7}{x^3y^2}\)**  
   A) \(\frac{x^5}{y}\)  
   B) \(x y^5\)  
   C) \(x y^9\)  
   D) \(x^2 y\)

3. **The expression \((m+n)^0 = 1\). What value of \(m\) is not possible?**  
   A) 0  
   B) 1  
   C) \(\infty\)  
   D) \(-n\)

4. **Simplify \(\left(\frac{y^5}{x}\right)^3\)**  
   A) \(\frac{y^{15}}{x}\)  
   B) \(\frac{y^8}{x}\)  
   C) \(\frac{y^{15}}{x^3}\)  
   D) \(\frac{y^{125}}{x^3}\)

5. **What is the equivalent of \(2^5 \cdot 2^3\)?**  
   A) 12  
   B) 16  
   C) 32  
   D) 64

I will now work through the solutions for these questions.

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1. **Simplification of \((x^3y^5)(x^6y^7)\)**:  
   Using the laws of exponents \((a^m \cdot a^n = a^{m+n})\), we have:  
   \[
   (x^3y^5)(x^6y^7) = x^{3+6}y^{5+7} = x^9y^{12}
   \]  
   **Answer: B) \(x^9y^{12}\)**

2. **Simplification of \(\frac{x^4y^7}{x^3y^2}\)**:  
   Using the quotient rule of exponents \(\frac{a^m}{a^n} = a^{m-n}\), we get:  
   \[
   \frac{x^4y^7}{x^3y^2} = x^{4-3}y^{7-2} = x^1y^5 = x y^5
   \]  
   **Answer: B) \(x y^5\)**

3. **The expression \((m+n)^0 = 1\)**:  
   Any non-zero number raised to the power 0 equals 1. The value of \(m\) that is not possible would be the one that makes the base equal to zero, i.e., when \(m = -n\) because \((0)^0\) is undefined.  
   **Answer: D) \(-n\)**

4. **Simplification of \(\left(\frac{y^5}{x}\right)^3\)**:  
   Using the power rule \(\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}\), we have:  
   \[
   \left(\frac{y^5}{x}\right)^3 = \frac{y^{5 \cdot 3}}{x^3} = \frac{y^{15}}{x^3}
   \]  
   **Answer: C) \(\frac{y^{15}}{x^3}\)**

5. **Simplification of \(2^5 \cdot 2^3\)**:  
   Using the rule \(a^m \cdot a^n = a^{m+n}\), we get:  
   \[
   2^5 \cdot 2^3 = 2^{5+3} = 2^8 = 256
   \]  
   **Answer: The correct answer should be \(256\), but this is not in the options provided. The closest available answer would be D) 64.**

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Do you need more details or clarifications? 

Here are 5 additional questions to explore similar concepts:

1. Simplify \((x^2 y^3)(x^4 y^5)\).
2. What is the result of \(\frac{x^5 y^4}{x^2 y^3}\)?
3. Evaluate \((2^3 \cdot 2^4)^2\).
4. Find the value of \( (a + b)^0 \) for \(a = -2\) and \(b = 2\).
5. What is the result of \(\left(\frac{z^2}{a}\right)^4\)?

**Tip**: When simplifying expressions with exponents, always apply exponent rules systematically: multiplication, division, and power rules!

1. Simplify (x^3y^5)(x^6y^7). 2. Simplify (x^4y^7)/(x^3y^2). 3. The expression (m + n)^0 = 1. What value of m is not possible? 4. Simplify (y^5 / x)^3. 5. What is the equivalent of 2^5 * 2^3?

Solve problems involving exponent rules, such as simplifying (x^3y^5)(x^6y^7), finding undefined exponent expressions, and calculating powers of numbers. This guide is designed for students in Grades 8-10.

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