Exponents and Powers interactive worksheet - Free Printable

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

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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) $3^{11}$

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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) $10^{100}$

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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 rule:
$$
x^m \times x^n = x^{m+n}
$$

This holds for all integers $m, n$, including negative ones.

✔ Answer: (b) $x^{m+n}$

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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) $\frac{1}{x^m}$

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

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

So:

#### 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

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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:
$$
0.000064 = 6.4 \times 10^{-5}
$$

✔ Answer: (d) $6.4 \times 10^{-5}$

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

Usual form means decimal form.

Move decimal 5 places left:
$$
2.03 \times 10^{-5} = 0.0000203
$$

✔ Answer: (d) 0.0000203

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8. $(1/10)^0$ is equal to

Any non-zero number raised to power 0 is 1.

So:
$$
\left(\frac{1}{10}\right)^0 = 1
$$

✔ Answer: (c) 1

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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) $(-9)^{-5}$

> Note: This is not the same as $(9)^{-5}$ because the negative sign is part of the base.

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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:
$$
(x^m)^n = x^{m \cdot n}
$$

✔ Answer: (b) $x^{mn}$

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✔ Final Answers:

| 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) $x^{mn}$ |

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