Grade 9 Exponent Rules Review Worksheets 2024 - Free Printable

Let's go through each problem on the Exponent Rules Review Worksheet and simplify them step by step using the exponent rules provided:

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🔷 Key Exponent 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 Power Rule: $ x^0 = 1 $ (for $ x \neq 0 $)
5. Negative Exponent Rule: $ x^{-n} = \frac{1}{x^n} $

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Now, let’s solve each problem:

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(1) $ a^2 \cdot a^3 $

- Same base → add exponents
- $ a^{2+3} = a^5 $

✔ Answer: $ a^5 $

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

- Multiply coefficients: $ 2 \cdot 4 = 8 $
- $ a^2 \cdot a = a^{2+1} = a^3 $
- $ b \cdot b^2 = b^{1+2} = b^3 $
- So: $ 8a^3b^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^{7} $
- So: $ -18x^7 $

✔ Answer: $ -18x^7 $

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(4) $ b^3 \cdot b^4 \cdot b^7 \cdot b $

- All bases are $ b $, so add all exponents:
- $ b^{3+4+7+1} = b^{15} $

✔ Answer: $ b^{15} $

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(5) $ (3x^3)(3x^4)(-3x^2) $

- Coefficients: $ 3 \cdot 3 \cdot (-3) = -27 $
- $ x^3 \cdot x^4 \cdot x^2 = x^{3+4+2} = x^9 $
- So: $ -27x^9 $

✔ Answer: $ -27x^9 $

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

- Apply power rule to each part:
- $ 2^2 = 4 $
- $ (x^2)^2 = x^{4} $
- $ (y^3)^2 = y^{6} $
- So: $ 4x^4y^6 $

✔ Answer: $ 4x^4y^6 $

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

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

✔ Answer: $ 125x^6y^{12} $

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(8) $ (6x^4y^6)^3 $

- $ 6^3 = 216 $
- $ (x^4)^3 = x^{12} $
- $ (y^6)^3 = y^{18} $
- So: $ 216x^{12}y^{18} $

✔ Answer: $ 216x^{12}y^{18} $

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

- $ 4^3 = 64 $
- $ (x^3)^3 = x^9 $
- $ (y^3)^3 = y^9 $
- So: $ 64x^9y^9 $

✔ Answer: $ 64x^9y^9 $

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(10) $ (7xy)^2 $

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

✔ Answer: $ 49x^2y^2 $

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(11) $ \frac{x^3}{x} $

- $ x^3 / x^1 = x^{3-1} = x^2 $

✔ Answer: $ x^2 $

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

- Coefficients: $ 18 / (-3) = -6 $
- $ c^3 / c^2 = c^{1} $
- So: $ -6c $

✔ Answer: $ -6c $

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

- Coefficients: $ 9 / (-3) = -3 $
- $ a^3 / a = a^{2} $
- $ b^5 / b^2 = b^{3} $
- So: $ -3a^2b^3 $

✔ Answer: $ -3a^2b^3 $

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

- Coefficients: $ (-48)/(-8) = 6 $
- $ c^2 / c = c^1 $
- $ d^4 / d = d^3 $
- So: $ 6cd^3 $

✔ Answer: $ 6cd^3 $

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(15) $ \frac{22y^6z^8}{2yz^{-7}} $

- Coefficients: $ 22 / 2 = 11 $
- $ y^6 / y = y^{5} $
- $ z^8 / z^{-7} = z^{8 - (-7)} = z^{15} $
- So: $ 11y^5z^{15} $

✔ Answer: $ 11y^5z^{15} $

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

- Different variables → already simplified
- No like terms

✔ Answer: $ x^2y^7 $

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(17) $ (x^2)^7 $

- Power rule: $ x^{2 \cdot 7} = x^{14} $

✔ Answer: $ x^{14} $

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(18) $ (-2x^4)^5 $

- $ (-2)^5 = -32 $
- $ (x^4)^5 = x^{20} $
- So: $ -32x^{20} $

✔ Answer: $ -32x^{20} $

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(19) $ 2x^3 + 7x^3 $

- Like terms: $ (2 + 7)x^3 = 9x^3 $

✔ Answer: $ 9x^3 $

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(20) $ 7^0 $

- Any non-zero number 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) $ -3^4 $

- This is not $ (-3)^4 $. It means $ -(3^4) $
- $ 3^4 = 81 $, so $ -81 $

⚠️ Note: The negative sign is not inside parentheses.

✔ Answer: $ -81 $

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(23) $ (-3)^4 $

- Negative number raised to even power → positive
- $ (-3)^4 = 81 $

✔ Answer: $ 81 $

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(24) $ 6x^0y^8 - (2y^2)^4 $

- $ x^0 = 1 $ → $ 6(1)y^8 = 6y^8 $
- $ (2y^2)^4 = 2^4 \cdot (y^2)^4 = 16y^8 $
- So: $ 6y^8 - 16y^8 = -10y^8 $

✔ Answer: $ -10y^8 $

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

- Difference of squares: $ a^2 - b^2 $
- $ = x^2 - (2y)^2 = x^2 - 4y^2 $

✔ Answer: $ x^2 - 4y^2 $

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

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

✔ Answer: $ -\frac{1}{4x} $

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(27) $ \frac{xy^7}{x^3y^4} $

- $ x / x^3 = x^{-2} $
- $ y^7 / y^4 = y^3 $
- So: $ x^{-2}y^3 = \frac{y^3}{x^2} $

✔ Answer: $ \frac{y^3}{x^2} $

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(28) $ 6x^5 \cdot 3x^5 \cdot x^0 $

- $ x^0 = 1 $
- Coefficients: $ 6 \cdot 3 = 18 $
- $ x^5 \cdot x^5 = x^{10} $
- So: $ 18x^{10} $

✔ Answer: $ 18x^{10} $

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(29) $ (3st^{12})^3 $

- $ 3^3 = 27 $
- $ s^3 $
- $ (t^{12})^3 = t^{36} $
- So: $ 27s^3t^{36} $

✔ Answer: $ 27s^3t^{36} $

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(30) $ \left( \frac{3m^2n^7}{m} \right)^5 $

- First simplify inside:
- $ \frac{3m^2n^7}{m} = 3m^{2-1}n^7 = 3mn^7 $
- Now raise to 5th power:
- $ (3mn^7)^5 = 3^5 \cdot m^5 \cdot (n^7)^5 = 243m^5n^{35} $

✔ Answer: $ 243m^5n^{35} $

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✔ Final Answers Summary:

| Problem | Answer |
|--------|--------|
| (1) | $ a^5 $ |
| (2) | $ 8a^3b^3 $ |
| (3) | $ -18x^7 $ |
| (4) | $ b^{15} $ |
| (5) | $ -27x^9 $ |
| (6) | $ 4x^4y^6 $ |
| (7) | $ 125x^6y^{12} $ |
| (8) | $ 216x^{12}y^{18} $ |
| (9) | $ 64x^9y^9 $ |
| (10) | $ 49x^2y^2 $ |
| (11) | $ x^2 $ |
| (12) | $ -6c $ |
| (13) | $ -3a^2b^3 $ |
| (14) | $ 6cd^3 $ |
| (15) | $ 11y^5z^{15} $ |
| (16) | $ x^2y^7 $ |
| (17) | $ x^{14} $ |
| (18) | $ -32x^{20} $ |
| (19) | $ 9x^3 $ |
| (20) | $ 1 $ |
| (21) | $ 8 $ |
| (22) | $ -81 $ |
| (23) | $ 81 $ |
| (24) | $ -10y^8 $ |
| (25) | $ x^2 - 4y^2 $ |
| (26) | $ -\frac{1}{4x} $ |
| (27) | $ \frac{y^3}{x^2} $ |
| (28) | $ 18x^{10} $ |
| (29) | $ 27s^3t^{36} $ |
| (30) | $ 243m^5n^{35} $ |

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