Exponents - Mixed Practice Worksheet 1: Simplify expressions and eliminate negative exponents in this algebra practice sheet.

Absolutely! Let’s solve each problem step by step, simplifying the expressions and ensuring no negative exponents remain in the final answer.

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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 for 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. \( (5p^4)^3 \)

Apply exponent to both coefficient and variable:
\( (ab)^n = a^n b^n \), and \( (p^a)^b = p^{a \cdot b} \)

→ \( 5^3 \cdot (p^4)^3 = 125 \cdot p^{12} = 125p^{12} \)

✔ Answer: \( \boxed{125p^{12}} \)

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

Apply exponent to each factor inside:

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

✔ Answer: \( \boxed{9x^{18}y^6} \)

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

Negative exponent means 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. \( (156u^{24}w^4)^0 \)

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:

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

✔ 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 x: \( x^{5+4} = x^9 \)
Add exponents for y: \( y^{1+7} = y^8 \)

→ \( 6x^9y^8 \)

✔ Answer: \( \boxed{6x^9y^8} \)

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

First simplify each part:

→ \( (2^3)(y^{4 \cdot 3}) = 8y^{12} \)
→ \( (5^2)(y^{6 \cdot 2}) = 25y^{12} \)

Now multiply:
→ \( 8 \cdot 25 \cdot y^{12+12} = 200y^{24} \)

✔ 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 apply exponent -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:

→ \( \frac{50}{2} = 25 \), \( x^{6-3} = x^3 \) → \( 25x^3 \)

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 each part:

→ \( (8k^{-7}j^{12})^2 = 8^2 \cdot k^{-14} \cdot j^{24} = 64k^{-14}j^{24} \)

Now multiply by \( 4k^{-4}j^3 \):

→ Coefficients: \( 64 \cdot 4 = 256 \)
k exponents: \( k^{-14 + (-4)} = k^{-18} \)
j exponents: \( j^{24+3} = j^{27} \)

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

✔ 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: \( 4b \cdot a^3b^7 = 4a^3b^8 \)

→ \( \frac{16a^{14}b^6}{4a^3b^8} = \frac{16}{4} \cdot a^{14-3} \cdot b^{6-8} = 4a^{11}b^{-2} \)

Convert to positive exponent:
→ \( \frac{4a^{11}}{b^2} \)

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

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

Simplify each fraction first:

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

Simplify each part:

→ \( (3a^2b^6)^2 = 9a^4b^{12} \)
→ \( (3a^{10}b^4)^3 = 27a^{30}b^{12} \)

Multiply:
→ \( 9 \cdot 27 = 243 \)
→ \( a^{4+30} = a^{34} \)
→ \( b^{12+12} = b^{24} \)

→ \( 243a^{34}b^{24} \)

✔ 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: \( \frac{5}{15} = \frac{1}{3} \)
a: \( a^{10 - (-2)} = a^{12} \)
b: \( b^{5 - 14} = b^{-9} \)
c: \( c^{22 - 11} = c^{11} \)

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

Now raise to power 3:

→ \( \left( \frac{1}{3} \right)^3 \cdot (a^{12})^3 \cdot (b^{-9})^3 \cdot (c^{11})^3 \)
→ \( \frac{1}{27} \cdot a^{36} \cdot b^{-27} \cdot c^{33} \)
→ \( \frac{a^{36}c^{33}}{27b^{27}} \)

✔ 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. \( 125p^{12} \)
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. \( \frac{a^{36}c^{33}}{27b^{27}} \)

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