Exponents - Riverside Math - Free Printable

Of course! Let’s solve each problem step by step, simplifying the expressions and ensuring no negative exponents appear in the final answers.

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1. $ x^4 \cdot x^7 $

Rule: When multiplying same bases, add exponents: $ x^a \cdot x^b = x^{a+b} $

→ $ x^{4+7} = x^{11} $

✔ Answer: $ \boxed{x^{11}} $

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2. $ 2x^6 \cdot x^{11} $

Multiply coefficients (only 2 here) and add exponents of $x$:

→ $ 2 \cdot x^{6+11} = 2x^{17} $

✔ Answer: $ \boxed{2x^{17}} $

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3. $ \frac{s^{14}}{s^8} $

Rule: When dividing same bases, subtract exponents: $ \frac{s^a}{s^b} = s^{a-b} $

→ $ s^{14-8} = s^6 $

✔ Answer: $ \boxed{s^6} $

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4. $ (5p^6)^3 $

Rule: Power of a product: $ (ab)^n = a^n b^n $

→ $ 5^3 \cdot (p^6)^3 = 125 \cdot p^{18} $

✔ Answer: $ \boxed{125p^{18}} $

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5. $ (3x^9y^3)^2 $

Apply exponent to each factor inside:

→ $ 3^2 \cdot (x^9)^2 \cdot (y^3)^2 = 9 \cdot x^{18} \cdot y^6 $

✔ Answer: $ \boxed{9x^{18}y^6} $

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6. $ (2m)^{-4} $

Negative exponent → take reciprocal:

→ $ \frac{1}{(2m)^4} = \frac{1}{2^4 m^4} = \frac{1}{16m^4} $

✔ Answer: $ \boxed{\frac{1}{16m^4}} $

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7. $ a^{-5} \cdot a^{-11} $

Add exponents:

→ $ a^{-5 + (-11)} = a^{-16} $

Convert to positive exponent:

→ $ \frac{1}{a^{16}} $

✔ Answer: $ \boxed{\frac{1}{a^{16}}} $

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8. $ \frac{25x^9}{5x^{12}} $

Simplify coefficients: $ \frac{25}{5} = 5 $

Subtract exponents: $ x^{9-12} = x^{-3} $

→ $ 5x^{-3} = \frac{5}{x^3} $

✔ Answer: $ \boxed{\frac{5}{x^3}} $

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9. $ (156u^{24}w^4)^0 $

Rule: Any non-zero expression to the power 0 is 1.

→ $ 1 $

✔ Answer: $ \boxed{1} $

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10. $ (5x^5y^4)^2 $

→ $ 5^2 \cdot (x^5)^2 \cdot (y^4)^2 = 25x^{10}y^8 $

✔ Answer: $ \boxed{25x^{10}y^8} $

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11. $ (-a^2b^5)^7 $

Apply exponent to each part, including the negative sign:

→ $ (-1)^7 \cdot (a^2)^7 \cdot (b^5)^7 = -1 \cdot a^{14} \cdot b^{35} $

✔ Answer: $ \boxed{-a^{14}b^{35}} $

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12. $ (2x^5y)(3x^4y^7) $

Multiply coefficients: $ 2 \cdot 3 = 6 $

Add exponents for like bases:

→ $ x^{5+4} = x^9 $, $ y^{1+7} = y^8 $

→ $ 6x^9y^8 $

✔ Answer: $ \boxed{6x^9y^8} $

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13. $ (2y^4)^3(5y^6)^2 $

First simplify each part:

→ $ (2^3)(y^{4\cdot3}) = 8y^{12} $

→ $ (5^2)(y^{6\cdot2}) = 25y^{12} $

Now multiply: $ 8 \cdot 25 = 200 $, $ y^{12+12} = y^{24} $

→ $ 200y^{24} $

✔ Answer: $ \boxed{200y^{24}} $

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14. $ \left( \frac{10x^9}{x^{18}} \right)^{-5} $

First simplify inside parentheses:

→ $ \frac{10x^9}{x^{18}} = 10x^{9-18} = 10x^{-9} $

Now raise to -5:

→ $ (10x^{-9})^{-5} = 10^{-5} \cdot x^{(-9)(-5)} = \frac{1}{10^5} \cdot x^{45} = \frac{x^{45}}{100000} $

✔ Answer: $ \boxed{\frac{x^{45}}{100000}} $

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15. $ \left( \frac{50x^6}{2x^3} \right)^2 $

Simplify inside first:

→ $ \frac{50}{2} = 25 $, $ x^{6-3} = x^3 $ → so $ 25x^3 $

Now square it:

→ $ (25x^3)^2 = 625x^6 $

✔ Answer: $ \boxed{625x^6} $

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16. $ (8k^{-7}j^{12})^2(4k^{-4}j^3) $

First simplify $ (8k^{-7}j^{12})^2 $:

→ $ 64k^{-14}j^{24} $

Now multiply by $ 4k^{-4}j^3 $:

→ Coefficients: $ 64 \cdot 4 = 256 $

Exponents: $ k^{-14 + (-4)} = k^{-18} $, $ j^{24+3} = j^{27} $

→ $ 256k^{-18}j^{27} = \frac{256j^{27}}{k^{18}} $

✔ Answer: $ \boxed{\frac{256j^{27}}{k^{18}}} $

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17. $ \frac{a^5b^2}{4b} \cdot \frac{16a^9b^4}{a^3b^7} $

Multiply numerators and denominators:

Numerator: $ a^5b^2 \cdot 16a^9b^4 = 16a^{14}b^6 $

Denominator: $ 4b \cdot a^3b^7 = 4a^3b^8 $

→ $ \frac{16a^{14}b^6}{4a^3b^8} $

Simplify:

Coefficients: $ \frac{16}{4} = 4 $

Exponents: $ a^{14-3} = a^{11} $, $ b^{6-8} = b^{-2} $

→ $ 4a^{11}b^{-2} = \frac{4a^{11}}{b^2} $

✔ Answer: $ \boxed{\frac{4a^{11}}{b^2}} $

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18. $ \frac{9x^7y^2}{xy} \cdot \frac{2x^6y^8}{2x^3y^4} $

First simplify each fraction:

First fraction: $ \frac{9x^7y^2}{xy} = 9x^{6}y $

Second fraction: $ \frac{2x^6y^8}{2x^3y^4} = x^{3}y^{4} $

Now multiply: $ 9x^6y \cdot x^3y^4 = 9x^{9}y^{5} $

✔ Answer: $ \boxed{9x^9y^5} $

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19. $ (3a^2b^6)^2(3a^{10}b^4)^3 $

First simplify each part:

→ $ (3a^2b^6)^2 = 9a^4b^{12} $

→ $ (3a^{10}b^4)^3 = 27a^{30}b^{12} $

Now multiply:

Coefficients: $ 9 \cdot 27 = 243 $

Exponents: $ a^{4+30} = a^{34} $, $ b^{12+12} = b^{24} $

→ $ 243a^{34}b^{24} $

✔ Answer: $ \boxed{243a^{34}b^{24}} $

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20. $ \left( \frac{5a^{10}b^5c^{22}}{15a^{-2}b^{14}c^{11}} \right)^3 $

First simplify inside the parentheses:

Coefficients: $ \frac{5}{15} = \frac{1}{3} $

Exponents:

- $ a^{10 - (-2)} = a^{12} $
- $ b^{5 - 14} = b^{-9} $
- $ c^{22 - 11} = c^{11} $

So inside: $ \frac{1}{3}a^{12}b^{-9}c^{11} $

Now cube everything:

→ $ \left( \frac{1}{3} \right)^3 \cdot a^{36} \cdot b^{-27} \cdot c^{33} = \frac{1}{27}a^{36}b^{-27}c^{33} $

Convert negative exponent:

→ $ \frac{a^{36}c^{33}}{27b^{27}} $

✔ Answer: $ \boxed{\frac{a^{36}c^{33}}{27b^{27}}} $

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## ✔ Final Answers Summary:

1. $ x^{11} $
2. $ 2x^{17} $
3. $ s^6 $
4. $ 125p^{18} $
5. $ 9x^{18}y^6 $
6. $ \frac{1}{16m^4} $
7. $ \frac{1}{a^{16}} $
8. $ \frac{5}{x^3} $
9. $ 1 $
10. $ 25x^{10}y^8 $
11. $ -a^{14}b^{35} $
12. $ 6x^9y^8 $
13. $ 200y^{24} $
14. $ \frac{x^{45}}{100000} $
15. $ 625x^6 $
16. $ \frac{256j^{27}}{k^{18}} $
17. $ \frac{4a^{11}}{b^2} $
18. $ 9x^9y^5 $
19. $ 243a^{34}b^{24} $
20. $ \frac{a^{36}c^{33}}{27b^{27}} $

Let me know if you’d like to see any step explained further!