Solved first 10 questionExponent Rules Review Worksheet | Chegg.com - Free Printable

Let's go through each problem in the Exponent Rules Review Worksheet step by step, applying the Product Rule, Power Rule, and Quotient Rule as needed.

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🔹 Key Rules Recap:

1. Product Rule: $ x^m \cdot x^n = x^{m+n} $
2. Power Rule: $ (x^m)^n = x^{m \cdot n} $
3. Quotient Rule: $ \frac{x^m}{x^n} = x^{m-n} $
4. Zero Exponent: $ x^0 = 1 $ (for $ x \neq 0 $)
5. Negative Exponent: $ x^{-n} = \frac{1}{x^n} $

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Now, let’s solve each problem:

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1) $ a \cdot a^2 \cdot a^4 $

Use Product Rule:
$$
a^1 \cdot a^2 \cdot a^4 = a^{1+2+4} = a^7
$$

✔ Answer: $ a^7 $

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2) $ (2a^2b)(4ab^2) $

Multiply coefficients and apply product rule for variables:
- Coefficients: $ 2 \cdot 4 = 8 $
- $ a^2 \cdot a = a^{2+1} = a^3 $
- $ b \cdot b^2 = b^{1+2} = b^3 $

✔ Answer: $ 8a^3b^3 $

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3) $ (6x^2)(-3x^5) $

- Coefficients: $ 6 \cdot (-3) = -18 $
- $ x^2 \cdot x^5 = x^{2+5} = x^7 $

✔ Answer: $ -18x^7 $

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4) $ b^3 \cdot b^1 \cdot b^1 \cdot b $

All are powers of $ b $:
$$
b^3 \cdot b \cdot b \cdot b = b^{3+1+1+1} = b^6
$$

✔ Answer: $ b^6 $

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5) $ (3x^3)(3x^3)(-3x^2) $

First multiply coefficients: $ 3 \cdot 3 \cdot (-3) = -27 $

Now powers of $ x $: $ x^3 \cdot x^3 \cdot x^2 = x^{3+3+2} = x^8 $

✔ Answer: $ -27x^8 $

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6) $ (2x^2)^3 y^3 $

Apply Power Rule to $ (2x^2)^3 $:
- $ 2^3 = 8 $
- $ (x^2)^3 = x^{2\cdot3} = x^6 $
So $ (2x^2)^3 = 8x^6 $

Then multiply by $ y^3 $: $ 8x^6y^3 $

✔ Answer: $ 8x^6y^3 $

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7) $ (5x^2y^4)^3 $

Apply Power Rule:
- $ 5^3 = 125 $
- $ (x^2)^3 = x^6 $
- $ (y^4)^3 = y^{12} $

✔ Answer: $ 125x^6y^{12} $

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8) $ (6x^3y^2)^3 $

- $ 6^3 = 216 $
- $ (x^3)^3 = x^9 $
- $ (y^2)^3 = y^6 $

✔ Answer: $ 216x^9y^6 $

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9) $ (4x^3y^2)^2 $

- $ 4^2 = 16 $
- $ (x^3)^2 = x^6 $
- $ (y^2)^2 = y^4 $

✔ Answer: $ 16x^6y^4 $

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10) $ (7xy)^2 $

- $ 7^2 = 49 $
- $ x^2 $
- $ y^2 $

✔ Answer: $ 49x^2y^2 $

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11) $ \frac{x^5}{x} $

Use Quotient Rule: $ x^{5-1} = x^4 $

✔ Answer: $ x^4 $

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12) $ \frac{18c^3}{-3c^2} $

- Coefficients: $ \frac{18}{-3} = -6 $
- $ c^{3-2} = c^1 = c $

✔ Answer: $ -6c $

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13) $ \frac{9a^3b^5}{-3ab^2} $

- Coefficients: $ \frac{9}{-3} = -3 $
- $ a^{3-1} = a^2 $
- $ b^{5-2} = b^3 $

✔ Answer: $ -3a^2b^3 $

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14) $ \frac{-48c^2d^4}{-8cd} $

- Coefficients: $ \frac{-48}{-8} = 6 $
- $ c^{2-1} = c^1 = c $
- $ d^{4-1} = d^3 $

✔ Answer: $ 6cd^3 $

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15) $ \frac{22y^5z^2}{2yz} $

- Coefficients: $ \frac{22}{2} = 11 $
- $ y^{5-1} = y^4 $
- $ z^{2-1} = z $

✔ Answer: $ 11y^4z $

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16) $ x^2 \cdot x^7 $

Product Rule: $ x^{2+7} = x^9 $

✔ Answer: $ x^9 $

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17) $ (x^3)^7 $

Power Rule: $ x^{3\cdot7} = x^{21} $

✔ Answer: $ x^{21} $

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18) $ (-2x)^3 $

Apply power to both coefficient and variable:
- $ (-2)^3 = -8 $
- $ x^3 $

✔ Answer: $ -8x^3 $

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19) $ 2x^2 + 7x^2 $

Combine like terms: $ (2+7)x^2 = 9x^2 $

✔ Answer: $ 9x^2 $

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20) $ 7^0 $

Anything to the zero power is 1.

✔ Answer: $ 1 $

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21) $ 8x^0 $

$ x^0 = 1 $, so $ 8 \cdot 1 = 8 $

✔ Answer: $ 8 $

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22) $ -3^2 $

⚠️ Important: This is not $ (-3)^2 $. It means $ -(3^2) = -9 $

✔ Answer: $ -9 $

> Note: If it were $ (-3)^2 $, it would be $ 9 $, but here it's just $ -3^2 $, so negative sign applies after squaring.

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23) $ (-3)^4 $

Even exponent → positive result:
$ (-3)^4 = 81 $

✔ Answer: $ 81 $

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24) $ 6x^8y^2 - (2y)^4 $

First simplify $ (2y)^4 = 2^4 \cdot y^4 = 16y^4 $

But we have $ 6x^8y^2 - 16y^4 $ — these are not like terms (different powers of $ x $ and $ y $), so cannot combine.

✔ Answer: $ 6x^8y^2 - 16y^4 $

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25) $ (x+2y)(x-2y) $

This is a difference of squares:
$$
(x+2y)(x-2y) = x^2 - (2y)^2 = x^2 - 4y^2
$$

✔ Answer: $ x^2 - 4y^2 $

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26) $ \frac{2x^4}{-8x^3} $

- Coefficients: $ \frac{2}{-8} = -\frac{1}{4} $
- $ x^{4-3} = x^1 = x $

So: $ -\frac{1}{4}x $

✔ Answer: $ -\frac{1}{4}x $

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27) $ \frac{3y^7}{x^3y^2} $

- $ y^{7-2} = y^5 $
- Denominator has $ x^3 $, so:
$$
\frac{3y^5}{x^3}
$$

✔ Answer: $ \frac{3y^5}{x^3} $

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28) $ 6x^2 \cdot 3x^2 \cdot x^6 $

Coefficients: $ 6 \cdot 3 = 18 $

Powers of $ x $: $ x^{2+2+6} = x^{10} $

✔ Answer: $ 18x^{10} $

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29) $ (3a^2)^3 $

- $ 3^3 = 27 $
- $ (a^2)^3 = a^6 $

✔ Answer: $ 27a^6 $

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30) $ \left( \frac{3m^2n^{-3}}{m} \right)^3 $

Simplify inside first:
$$
\frac{3m^2n^{-3}}{m} = 3m^{2-1}n^{-3} = 3mn^{-3}
$$

Now raise to the 3rd power:
$$
(3mn^{-3})^3 = 3^3 \cdot m^3 \cdot (n^{-3})^3 = 27m^3n^{-9}
$$

We can write with positive exponents:
$$
27m^3 \cdot \frac{1}{n^9} = \frac{27m^3}{n^9}
$$

✔ Answer: $ \frac{27m^3}{n^9} $

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✔ Final Answers Summary:

| Problem | Answer |
|--------|--------|
| 1 | $ a^7 $ |
| 2 | $ 8a^3b^3 $ |
| 3 | $ -18x^7 $ |
| 4 | $ b^6 $ |
| 5 | $ -27x^8 $ |
| 6 | $ 8x^6y^3 $ |
| 7 | $ 125x^6y^{12} $ |
| 8 | $ 216x^9y^6 $ |
| 9 | $ 16x^6y^4 $ |
| 10 | $ 49x^2y^2 $ |
| 11 | $ x^4 $ |
| 12 | $ -6c $ |
| 13 | $ -3a^2b^3 $ |
| 14 | $ 6cd^3 $ |
| 15 | $ 11y^4z $ |
| 16 | $ x^9 $ |
| 17 | $ x^{21} $ |
| 18 | $ -8x^3 $ |
| 19 | $ 9x^2 $ |
| 20 | $ 1 $ |
| 21 | $ 8 $ |
| 22 | $ -9 $ |
| 23 | $ 81 $ |
| 24 | $ 6x^8y^2 - 16y^4 $ |
| 25 | $ x^2 - 4y^2 $ |
| 26 | $ -\frac{1}{4}x $ |
| 27 | $ \frac{3y^5}{x^3} $ |
| 28 | $ 18x^{10} $ |
| 29 | $ 27a^6 $ |
| 30 | $ \frac{27m^3}{n^9} $ |

Let me know if you'd like this formatted as a printable worksheet or need explanations for any specific one!