Laws of 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:

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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 is $ 3x^4 $, so:
- $ (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} $

But wait — if the problem wants us to use the product rule, we might treat this as:
$$
(3x^4)^2 \cdot (3x^4)^3 = (3x^4)^{2+3} = (3x^4)^5
$$

So two possible answers depending on interpretation.

But since it says "use the product rule", and the base is the same ($ 3x^4 $), then:
$$
(3x^4)^2 \cdot (3x^4)^3 = (3x^4)^{5}
$$

✔ Answer: $ (3x^4)^5 $

(Note: If expanded: $ 3^5 \cdot x^{20} = 243x^{20} $, but unless asked to expand, leave as $ (3x^4)^5 $)

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#### 12. $ x^3 \cdot x^7 \cdot x^2 $
- $ x^{3+7+2} = x^{12} $

✔ Answer: $ x^{12} $

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(b) Find the value of:

Use the product rule first, then evaluate.

#### 1. $ 10^3 \cdot 10^4 \cdot 10^2 $
- $ 10^{3+4+2} = 10^9 $
- $ 10^9 = 1,000,000,000 $

✔ Answer: $ 1,000,000,000 $

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#### 2. $ (-4)^2 \cdot (-4)^3 \cdot (-4)^2 $
- $ (-4)^{2+3+2} = (-4)^7 $
- $ (-4)^7 = -16384 $ (since odd power of negative number is negative)
- $ 4^7 = 16384 $, so $ (-4)^7 = -16384 $

✔ Answer: $ -16384 $

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#### 3. $ 6^2 \cdot 6^2 \cdot 6^3 \cdot 6^3 \cdot 6^2 $
- Add exponents: $ 2+2+3+3+2 = 12 $
- $ 6^{12} $ → large number, but just leave as $ 6^{12} $ unless asked to compute.

✔ Answer: $ 6^{12} $

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#### 4. $ (2x^3)^2 \cdot (2x^3)^3 $
- Same base: $ (2x^3)^{2+3} = (2x^3)^5 $
- Expand: $ 2^5 \cdot (x^3)^5 = 32x^{15} $

✔ Answer: $ 32x^{15} $

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#### 5. $ (1000)^2 \cdot (1000)^3 \cdot (1000)^2 $
- $ 1000 = 10^3 $, so:
- $ (10^3)^2 = 10^6 $
- $ (10^3)^3 = 10^9 $
- $ (10^3)^2 = 10^6 $
- Multiply: $ 10^6 \cdot 10^9 \cdot 10^6 = 10^{6+9+6} = 10^{21} $

Alternatively: $ (1000)^{2+3+2} = (1000)^7 = (10^3)^7 = 10^{21} $

✔ Answer: $ 10^{21} $

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

✔ Answer: $ x^{10} $

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(c) Which of the following equals $ 2 \cdot 10^4 \cdot 3 \cdot 10^6 $?

We have:
$ 2 \cdot 10^4 \cdot 3 \cdot 10^6 = (2 \cdot 3) \cdot (10^4 \cdot 10^6) = 6 \cdot 10^{10} $

Now check options:

(a) $ 6 \cdot 10^{10} $ ✔ Correct

(b) $ 5 \cdot 10^{10} $ ✘

(c) $ 10^{10} $ ✘

(d) $ 6 \cdot 10^6 $ ✘

✔ Answer: (a)

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(d) Find the value of: $ 3 \cdot 10^4 \cdot 2 \cdot 10^3 \cdot 4 \cdot 10^5 $

Group constants and powers of 10:

- Constants: $ 3 \cdot 2 \cdot 4 = 24 $
- Powers of 10: $ 10^4 \cdot 10^3 \cdot 10^5 = 10^{4+3+5} = 10^{12} $
- So total: $ 24 \cdot 10^{12} $

But in scientific notation, we write it as $ 2.4 \cdot 10^{13} $

Because $ 24 \cdot 10^{12} = 2.4 \cdot 10^1 \cdot 10^{12} = 2.4 \cdot 10^{13} $

✔ Answer: $ 2.4 \times 10^{13} $

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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. $ (3x^4)^5 $
12. $ x^{12} $

(b)
1. $ 1,000,000,000 $
2. $ -16384 $
3. $ 6^{12} $
4. $ 32x^{15} $
5. $ 10^{21} $
6. $ x^{10} $

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

(d) $ \boxed{2.4 \times 10^{13}} $

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Let me know if you'd like these explained further or if there’s a part you're unsure about!