Exponents and Radicals - Integers with Simple Exponents Worksheets - Free Printable

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:

---

(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} $

---

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

✔ Answer: $ 10^{14} $

---

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

✔ Answer: $ a^{11} $

---

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

✔ Answer: $ (-4)^5 $

---

#### 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 $

---

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

✔ Answer: $ a^{15} $

---

#### 7. $ y^{10} \cdot y^{10} $
- $ y^{10+10} = y^{20} $

✔ Answer: $ y^{20} $

---

#### 8. $ 3^6 \cdot 3^3 \cdot 3^2 $
- $ 3^{6+3+2} = 3^{11} $

✔ Answer: $ 3^{11} $

---

#### 9. $ (-5)^4 \cdot (-5)^3 \cdot (-5)^2 $
- $ (-5)^{4+3+2} = (-5)^9 $

✔ Answer: $ (-5)^9 $

---

#### 10. $ x^2 \cdot x^5 \cdot x^3 \cdot x^7 $
- $ x^{2+5+3+7} = x^{17} $

✔ Answer: $ x^{17} $

---

#### 11. $ 2x^2 \cdot 3x^3 $
- First, multiply coefficients: $ 2 \cdot 3 = 6 $
- Then apply exponent rule: $ x^2 \cdot x^3 = x^{5} $
- So: $ 6x^5 $

✔ Answer: $ 6x^5 $

---

#### 12. $ (-3y^4)(-2y^5) $
- Multiply coefficients: $ (-3) \cdot (-2) = 6 $
- Multiply variables: $ y^4 \cdot y^5 = y^{9} $
- So: $ 6y^9 $

✔ Answer: $ 6y^9 $

---

#### 13. $ (-3x^2)(-4x^3) $
- Coefficients: $ (-3) \cdot (-4) = 12 $
- Variables: $ x^2 \cdot x^3 = x^5 $
- So: $ 12x^5 $

✔ Answer: $ 12x^5 $

---

#### 14. $ 2x^3 \cdot 3x^4 $
- $ 2 \cdot 3 = 6 $
- $ x^3 \cdot x^4 = x^7 $
- So: $ 6x^7 $

✔ Answer: $ 6x^7 $

---

#### 15. $ (-5y^3)(2y^2) $
- Coefficients: $ (-5) \cdot 2 = -10 $
- Variables: $ y^3 \cdot y^2 = y^5 $
- So: $ -10y^5 $

✔ Answer: $ -10y^5 $

---

#### 16. $ 3x^2 \cdot 4x^3 \cdot 2x^4 $
- Coefficients: $ 3 \cdot 4 \cdot 2 = 24 $
- Exponents: $ x^{2+3+4} = x^9 $
- So: $ 24x^9 $

✔ Answer: $ 24x^9 $

---

(b) Find the value of:

#### 1. $ 3^2 \cdot 3^4 = 3^{2+4} = 3^6 = 729 $

✔ Answer: $ 729 $

---

#### 2. $ 4^2 \cdot 4^3 = 4^{5} = 1024 $

✔ Answer: $ 1024 $

---

#### 3. $ 2^3 \cdot 2^5 = 2^8 = 256 $

✔ Answer: $ 256 $

---

#### 4. $ 5^2 \cdot 5^3 = 5^5 = 3125 $

✔ Answer: $ 3125 $

---

#### 5. $ (-2)^3 \cdot (-2)^4 = (-2)^7 = -128 $

Note: $ (-2)^7 = -128 $ because odd power of negative number is negative.

✔ Answer: $ -128 $

---

#### 6. $ (-3)^2 \cdot (-3)^3 = (-3)^5 = -243 $

✔ Answer: $ -243 $

---

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

We are given:
- a) $ 2 \cdot 10^6 $
- b) $ 2 \cdot 10^6 $
- c) $ 2 \cdot 10^6 $
- d) $ 2 \cdot 10^6 $

Wait — all options seem identical? That might be a typo in the original image.

But if the question is asking which one equals $ 2 \cdot 10^6 $, then all of them do, assuming they're written correctly.

Possibility: Maybe the options were meant to be different?

For example, perhaps:
- a) $ 2 \cdot 10^5 $
- b) $ 2 \cdot 10^6 $
- c) $ 20 \cdot 10^5 $
- d) $ 200 \cdot 10^4 $

But based on what you wrote, all options are $ 2 \cdot 10^6 $, so all are correct.

✔ Answer: All options (a, b, c, d) equal $ 2 \cdot 10^6 $ — unless there was a formatting error.

---

(d) Find the value of: $ 2 \cdot 10^3 \cdot 3 \cdot 10^4 $

Step-by-step:
- Group constants: $ 2 \cdot 3 = 6 $
- Group powers of 10: $ 10^3 \cdot 10^4 = 10^{3+4} = 10^7 $
- So: $ 6 \cdot 10^7 $

✔ Answer: $ 6 \cdot 10^7 $

---

✔ Final Summary of Answers:

#### (a)
1. $ x^{35} $
2. $ 10^{14} $
3. $ a^{11} $
4. $ (-4)^5 $
5. $ x^9 $
6. $ a^{15} $
7. $ y^{20} $
8. $ 3^{11} $
9. $ (-5)^9 $
10. $ x^{17} $
11. $ 6x^5 $
12. $ 6y^9 $
13. $ 12x^5 $
14. $ 6x^7 $
15. $ -10y^5 $
16. $ 24x^9 $

#### (b)
1. $ 729 $
2. $ 1024 $
3. $ 256 $
4. $ 3125 $
5. $ -128 $
6. $ -243 $

#### (c)
All options are $ 2 \cdot 10^6 $ → All are correct (unless miswritten)

#### (d)
$ 6 \cdot 10^7 $

---

If you can provide the exact wording or clarify any part (especially section (c)), I can adjust accordingly. Let me know!