Practice your algebra skills with this multiple-choice quiz on the laws of exponents and powers.

Let's solve each question from the Exponents and Powers – Worksheet step by step with explanations.

---

1. The value of $ 3^5 \div 3^{-6} $ is

We use the rule:
$$
\frac{a^m}{a^n} = a^{m - n}
$$

So:
$$
3^5 \div 3^{-6} = 3^{5 - (-6)} = 3^{5 + 6} = 3^{11}
$$

✔ Answer: (c) $ 3^{11} $

---

2. The multiplicative inverse of $ 10^{-100} $ is

The multiplicative inverse of a number $ a $ is $ \frac{1}{a} $.
So:
$$
\text{Inverse of } 10^{-100} = \frac{1}{10^{-100}} = 10^{100}
$$

✔ Answer: (c) $ 10^{100} $

---

3. If $ x $ be any non-zero integer and $ m, n $ be negative integers, then $ x^m \times x^n $ is equal to:

Use the law:
$$
x^m \times x^n = x^{m+n}
$$

This holds regardless of whether $ m $ and $ n $ are positive or negative.

✔ Answer: (b) $ x^{m+n} $

---

4. If $ x $ be any integer different from zero and $ m $ be any positive integer, then $ x^{-m} $ is equal to:

By definition:
$$
x^{-m} = \frac{1}{x^m}
$$

✔ Answer: (c) $ \frac{1}{x^m} $

---

5. The value of $ (7^{-1} - 8^{-1})^{-1} - (3^{-1} - 4^{-1})^{-1} $ is:

First, recall that $ a^{-1} = \frac{1}{a} $

So:
- $ 7^{-1} = \frac{1}{7},\quad 8^{-1} = \frac{1}{8} $
- $ 3^{-1} = \frac{1}{3},\quad 4^{-1} = \frac{1}{4} $

Now compute:

#### First term:
$$
(7^{-1} - 8^{-1})^{-1} = \left( \frac{1}{7} - \frac{1}{8} \right)^{-1}
= \left( \frac{8 - 7}{56} \right)^{-1} = \left( \frac{1}{56} \right)^{-1} = 56
$$

#### Second term:
$$
(3^{-1} - 4^{-1})^{-1} = \left( \frac{1}{3} - \frac{1}{4} \right)^{-1}
= \left( \frac{4 - 3}{12} \right)^{-1} = \left( \frac{1}{12} \right)^{-1} = 12
$$

Now subtract:
$$
56 - 12 = 44
$$

✔ Answer: (a) 44

---

6. The standard form for 0.000064 is

Standard form means scientific notation: $ a \times 10^n $, where $ 1 \leq a < 10 $

Move decimal point 5 places right to get 6.4:
$$
0.000064 = 6.4 \times 10^{-5}
$$

✔ Answer: (d) $ 6.4 \times 10^{-5} $

---

7. The usual form for $ 2.03 \times 10^{-5} $

$ 10^{-5} = 0.00001 $, so:
$$
2.03 \times 10^{-5} = 2.03 \times 0.00001 = 0.0000203
$$

✔ Answer: (d) 0.0000203

---

8. $ \left(\frac{1}{10}\right)^0 $ is equal to

Any non-zero number raised to power 0 is 1:
$$
\left(\frac{1}{10}\right)^0 = 1
$$

✔ Answer: (c) 1

---

9. $ (-9)^3 \div (-9)^8 $ is equal to:

Use the rule:
$$
\frac{a^m}{a^n} = a^{m-n}
$$

So:
$$
(-9)^3 \div (-9)^8 = (-9)^{3-8} = (-9)^{-5}
$$

✔ Answer: (d) $ (-9)^{-5} $

---

10. If $ x $ be any integer different from zero and $ m, n $ be any integers, then $ (x^m)^n $ is equal to:

Use the power of a power rule:
$$
(x^m)^n = x^{m \cdot n} = x^{mn}
$$

✔ Answer: (b) $ x^{mn} $

---

✔ Final Answers Summary:

| Question | Answer |
|--------|--------|
| 1 | (c) $ 3^{11} $ |
| 2 | (c) $ 10^{100} $ |
| 3 | (b) $ x^{m+n} $ |
| 4 | (c) $ \frac{1}{x^m} $ |
| 5 | (a) 44 |
| 6 | (d) $ 6.4 \times 10^{-5} $ |
| 7 | (d) 0.0000203 |
| 8 | (c) 1 |
| 9 | (d) $ (-9)^{-5} $ |
| 10 | (b) $ x^{mn} $ |

Let me know if you'd like this as a printable PDF or need further explanation on any step!