Laws of Exponents Worksheets - Free Printable

It looks like you've uploaded a worksheet titled "Exponents: Product Rule", which contains several problems related to applying the product rule of exponents. The product rule states:

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

This means when multiplying two powers with the same base, you add the exponents.

Let’s go through each problem step by step and solve them.

---

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

#### 1) $ x^{10} \cdot x^{25} $
- Same base: $ x $
- Add exponents: $ 10 + 25 = 35 $
- Answer: $ x^{35} $

#### 2) $ (10^3) \cdot (10^7) $
- Same base: $ 10 $
- Add exponents: $ 3 + 7 = 10 $
- Answer: $ 10^{10} $

#### 3) $ x^5 \cdot x^8 $
- Same base: $ x $
- Add exponents: $ 5 + 8 = 13 $
- Answer: $ x^{13} $

#### 4) $ (-6)^4 \cdot (-6)^2 $
- Same base: $ -6 $
- Add exponents: $ 4 + 2 = 6 $
- Answer: $ (-6)^6 $

#### 5) $ a \cdot a^3 \cdot a^4 $
- All terms have base $ a $. Remember $ a = a^1 $
- Exponents: $ 1 + 3 + 4 = 8 $
- Answer: $ a^8 $

#### 6) $ x^9 \cdot x^4 $
- Same base: $ x $
- Add exponents: $ 9 + 4 = 13 $
- Answer: $ x^{13} $

---

(b) Find the value of:

We now evaluate expressions numerically.

#### 1) $ 10^3 \cdot 10^4 = 10^{3+4} = 10^7 $
- $ 10^7 = 10,000,000 $
- Answer: $ 10,000,000 $

#### 2) $ (-6)^2 \cdot (-6)^3 = (-6)^{2+3} = (-6)^5 $
- $ (-6)^5 = -6 \cdot -6 \cdot -6 \cdot -6 \cdot -6 $
- Even number of negatives → positive? No: 5 is odd
- $ = -7776 $
- Answer: $ -7776 $

#### 3) $ 3x^3 \cdot 5x^2 = (3 \cdot 5)(x^3 \cdot x^2) = 15x^{3+2} = 15x^5 $
- Answer: $ 15x^5 $

#### 4) $ 2x^2 \cdot 3x^4 = (2 \cdot 3)(x^2 \cdot x^4) = 6x^{6} $
- Answer: $ 6x^6 $

#### 5) $ (-3x)^2 \cdot (-3x)^3 = (-3x)^{2+3} = (-3x)^5 $
- $ = (-3)^5 \cdot x^5 = -243x^5 $
- Answer: $ -243x^5 $

#### 6) $ (-2x^2)^3 \cdot (-2x^2)^2 = (-2x^2)^{3+2} = (-2x^2)^5 $
- $ = (-2)^5 \cdot (x^2)^5 = -32 \cdot x^{10} = -32x^{10} $
- Answer: $ -32x^{10} $

---

(c) Which of the following equals $ 2 \cdot 10^4 \cdot 3 \cdot 10^5 $?

We simplify:
- $ 2 \cdot 3 = 6 $
- $ 10^4 \cdot 10^5 = 10^{4+5} = 10^9 $
- So total: $ 6 \cdot 10^9 $

Now check options:

- a) $ 5 \cdot 10^9 $ → ✘
- b) $ 6 \cdot 10^9 $ → ✔
- c) $ 6 \cdot 10^5 $ → ✘
- d) $ 5 \cdot 10^5 $ → ✘

✔ Correct answer: b) $ 6 \cdot 10^9 $

---

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

From above: $ 6 \cdot 10^9 = 6,000,000,000 $

✔ Answer: $ 6,000,000,000 $

---

✔ Final Answers Summary:

#### (a)
1. $ x^{35} $
2. $ 10^{10} $
3. $ x^{13} $
4. $ (-6)^6 $
5. $ a^8 $
6. $ x^{13} $

#### (b)
1. $ 10,000,000 $
2. $ -7776 $
3. $ 15x^5 $
4. $ 6x^6 $
5. $ -243x^5 $
6. $ -32x^{10} $

#### (c) b) $ 6 \cdot 10^9 $

#### (d) $ 6,000,000,000 $

---

Let me know if you'd like this formatted as a printable answer sheet or need help understanding any step!