https://file.aiquickdraw.com/mathbot/file/1729769611330nt97jyfk.jpg

Here’s the breakdown for **Question 1** from the uploaded image.

These questions are based on **index notation rules**. Specifically:

1. **When multiplying terms with the same base, add their exponents**:
   \[ a^m \times a^n = a^{m+n} \]

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### Solutions:

**(a)** \( x^6 \times x^4 \)  
- Add the exponents: \( 6 + 4 = 10 \)  
  **Answer:** \( x^{10} \)

**(d)** \( y \times y^6 \)  
- Remember that \( y = y^1 \), so: \( 1 + 6 = 7 \)  
  **Answer:** \( y^7 \)

**(g)** \( 4x^5 \times 7x^3 \)  
- Multiply the constants: \( 4 \times 7 = 28 \)  
- Add the exponents for \(x\): \( 5 + 3 = 8 \)  
  **Answer:** \( 28x^8 \)

**(j)** \( m^7 \times m^5 \times m^4 \)  
- Add the exponents: \( 7 + 5 + 4 = 16 \)  
  **Answer:** \( m^{16} \)

**(l)** \( 5xy \times 4x^2y^3 \times 3x^3y^3 \)  
- Multiply the constants: \( 5 \times 4 \times 3 = 60 \)  
- Add the exponents for \(x\): \( 1 + 2 + 3 = 6 \)  
- Add the exponents for \(y\): \( 1 + 3 + 3 = 7 \)  
  **Answer:** \( 60x^6y^7 \)

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Would you like any of these steps further explained? 

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Here are 5 related questions to reinforce your learning:
1. What happens if you divide terms with the same base?
2. How would you simplify \( (x^3)^4 \)?
3. How does the rule change if the bases are different?
4. What is the result of multiplying \( 0 \times x^5 \)?
5. Can you simplify \( a^{-3} \times a^5 \)?

**Tip:** Always ensure the bases are identical before applying exponent rules!

Simplify the following algebraic expressions involving index notation.

Learn how to simplify algebraic expressions using index notation and exponent rules. This tutorial covers multiplying terms with the same base and adding exponents, with examples such as x^6 × x^4, y × y^6, and more. Suitable for students in Grades 7-9.

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