Exponents - Product Rule worksheet for practicing simplifying exponential expressions.

Since I can't view or access images directly, I can't see the specific content of your uploaded image. However, based on your description and the text you've provided, it appears to be a worksheet titled "Exponents - Product Rule" with several problems involving exponent rules, particularly the product rule:

> $$
> a^m \cdot a^n = a^{m+n}
> $$

Let me help you solve the problems based on the text you've typed. Here's the breakdown of each section:

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(a) Use the product rule to write each expression as a single exponent.

#### 1. $ x^{10} \cdot x^{25} $
- Apply the product rule: $ x^{10+25} = x^{35} $

✔ Answer: $ x^{35} $

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#### 2. $ (10^7) \cdot (10^7) $
- $ 10^7 \cdot 10^7 = 10^{7+7} = 10^{14} $

✔ Answer: $ 10^{14} $

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#### 3. $ a^3 \cdot a^8 $
- $ a^{3+8} = a^{11} $

✔ Answer: $ a^{11} $

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#### 4. $ (-4)^2 \cdot (-4)^3 $
- $ (-4)^{2+3} = (-4)^5 $

✔ Answer: $ (-4)^5 $

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#### 5. $ x \cdot x^2 \cdot x^6 $
- $ x^1 \cdot x^2 \cdot x^6 = x^{1+2+6} = x^9 $

✔ Answer: $ x^9 $

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#### 6. $ a^{10} \cdot a^5 $
- $ a^{10+5} = a^{15} $

✔ Answer: $ a^{15} $

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#### 7. $ y^6 \cdot y^2 $
- $ y^{6+2} = y^8 $

✔ Answer: $ y^8 $

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#### 8. $ 6^2 \cdot 6^5 $
- $ 6^{2+5} = 6^7 $

✔ Answer: $ 6^7 $

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#### 9. $ (x^2)^4 \cdot (x^2)^5 $
- First simplify powers: $ (x^2)^4 = x^{8} $, $ (x^2)^5 = x^{10} $
- Then multiply: $ x^8 \cdot x^{10} = x^{18} $

✔ Answer: $ x^{18} $

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#### 10. $ (x^3)^2 \cdot (x^3)^4 $
- $ (x^3)^2 = x^6 $, $ (x^3)^4 = x^{12} $
- $ x^6 \cdot x^{12} = x^{18} $

✔ Answer: $ x^{18} $

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#### 11. $ (3x^4)^2 \cdot (3x^4)^3 $
- This is tricky — the base includes a coefficient.
- $ (3x^4)^2 = 3^2 \cdot (x^4)^2 = 9x^8 $
- $ (3x^4)^3 = 27x^{12} $
- Now multiply: $ 9x^8 \cdot 27x^{12} = (9 \cdot 27)x^{8+12} = 243x^{20} $

✔ Answer: $ 243x^{20} $

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#### 12. $ x^5 \cdot x^6 \cdot x^3 $
- $ x^{5+6+3} = x^{14} $

✔ Answer: $ x^{14} $

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(b) Find the value of:

#### 1. $ x^3 \cdot x^4 \cdot x^5 $
- $ x^{3+4+5} = x^{12} $

✔ Answer: $ x^{12} $

#### 2. $ (-4)^2 \cdot (-4)^3 \cdot (-4)^4 $
- $ (-4)^{2+3+4} = (-4)^9 $

✔ Answer: $ (-4)^9 $

#### 3. $ 2x^4 \cdot 3x^3 \cdot 5x^2 $
- Multiply coefficients: $ 2 \cdot 3 \cdot 5 = 30 $
- Add exponents: $ x^{4+3+2} = x^9 $
- So: $ 30x^9 $

✔ Answer: $ 30x^9 $

#### 4. $ x^3 \cdot x^5 \cdot x^7 $
- $ x^{3+5+7} = x^{15} $

✔ Answer: $ x^{15} $

#### 5. $ 3x^3 \cdot 2x^5 \cdot 4x^8 $
- Coefficients: $ 3 \cdot 2 \cdot 4 = 24 $
- Exponents: $ x^{3+5+8} = x^{16} $
- Answer: $ 24x^{16} $

✔ Answer: $ 24x^{16} $

#### 6. $ x^7 \cdot x^3 $
- $ x^{7+3} = x^{10} $

✔ Answer: $ x^{10} $

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(c) Which of the following equals $ 2 \cdot 10^6 \cdot 3 \cdot 10^5 $?

We compute:
- $ 2 \cdot 3 = 6 $
- $ 10^6 \cdot 10^5 = 10^{11} $
- So total: $ 6 \cdot 10^{11} $

Now check options:

(a) $ 6 \cdot 10^{11} $ → ✔ Correct
(b) $ 6 \cdot 10^{10} $ → ✘
(c) $ 5 \cdot 10^{11} $ → ✘
(d) $ 5 \cdot 10^{10} $ → ✘

✔ Answer: (a) $ 6 \cdot 10^{11} $

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(d) Find the value of $ x $, if $ 2x^3 \cdot 3x^4 \cdot 5x^5 = 30x^k $.

First, simplify the left-hand side:
- Coefficients: $ 2 \cdot 3 \cdot 5 = 30 $
- Variables: $ x^3 \cdot x^4 \cdot x^5 = x^{3+4+5} = x^{12} $
- So: $ 30x^{12} $

Given: $ 30x^k $, so $ k = 12 $

✔ Answer: $ k = 12 $

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✔ Final Answers Summary:

(a)
1. $ x^{35} $
2. $ 10^{14} $
3. $ a^{11} $
4. $ (-4)^5 $
5. $ x^9 $
6. $ a^{15} $
7. $ y^8 $
8. $ 6^7 $
9. $ x^{18} $
10. $ x^{18} $
11. $ 243x^{20} $
12. $ x^{14} $

(b)
1. $ x^{12} $
2. $ (-4)^9 $
3. $ 30x^9 $
4. $ x^{15} $
5. $ 24x^{16} $
6. $ x^{10} $

(c) $ \boxed{\text{(a) } 6 \cdot 10^{11}} $

(d) $ \boxed{k = 12} $

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Let me know if you'd like this formatted as a printable answer sheet or need further explanation!