Simplify expressions using division properties of exponents.
Practice worksheet on division properties of exponents with problems to simplify expressions and quotients, including scientific notation.
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Step-by-step solution for: Solved Practice Division Properties of Exponents Simplify | Chegg.com
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Show Answer Key & Explanations
Step-by-step solution for: Solved Practice Division Properties of Exponents Simplify | Chegg.com
Final Answer:
1. $5^3$
2. $5^3$
3. $x^{\frac{5}{8}}$
4. $m^2$
5. $x^4 y^4$
6. $7m^{\frac{11}{12}}$
7. $\frac{81}{625}$
8. $\frac{27x^3}{8y^3}$
9. $\frac{49}{16}$
10. $-\frac{8y^{15}}{27x^{12}}$
11. $\frac{625p^4}{2304d^{12}}$
12. $\frac{a^4 b^4}{b^4} = a^4$ → actually: $\left(\frac{ab^2}{a^5 b}\right)^{-2} = \left(a^{-4}b\right)^{-2} = a^8 b^{-2} = \frac{a^8}{b^2}$
13. $\frac{125x^6 z^{15}}{27y^{15}}$
14. $-\frac{12n^8}{m^6}$
15. Not simplest: exponent on 2 is positive, but variable part $r^3$ has no coefficient — actually, it *is* simplest unless they expect no variables in denominator or something; but per standard, $2^4 r^3 = 16r^3$, so not simplest because $2^4$ should be evaluated.
16. Not simplest: $(3x)^2 = 9x^2$, so should be simplified to $9x^2$.
17. Not simplest: $n^0 = 1$, so $m^3 n^0 = m^3$.
18. Not simplest: $\frac{y^5}{y} = y^4$.
19. $2.4 \times 10^4$
20. $9 \times 10^{-5}$
But since the instruction says ExplainAnswer: false, TeachTopic: false, GeneratePractice: false, only the final answer section is allowed — and it must contain only the final correct answer, with no extra text.
However, the problem asks to simplify *each* expression, and there are 20 items, so the final answer must list all 20 simplified results clearly.
Let me recompute carefully and give exact simplified forms:
1. $\frac{5^6}{5^2} = 5^{6-2} = 5^4$ → Wait! Mistake earlier. $6-2=4$, so $5^4$, not $5^3$.
Let me redo all accurately:
1. $\frac{5^6}{5^2} = 5^{6-2} = 5^4$
2. $\frac{5^5}{5^2} = 5^{3} = 125$ or $5^3$ — keep as $5^3$ (exponential form preferred)
3. $\frac{x^{5/6}}{x^{1/8}} = x^{5/6 - 1/8} = x^{(20-3)/24} = x^{17/24}$
Wait — original says $x^{\frac{5}{6}} / x^{\frac{1}{8}}$? Yes. So exponent: $\frac{5}{6} - \frac{1}{8} = \frac{20-3}{24} = \frac{17}{24}$. So $x^{17/24}$
I previously wrote $x^{5/8}$ — wrong.
This shows I must be precise.
Given the complexity and risk of error in mental calculation for 20 items, and the instruction says *only final answer*, I will provide the correct simplified forms, double-checked:
1. $5^{4}$
2. $5^{3}$
3. $x^{\frac{17}{24}}$
4. $\frac{m^{-3}}{m^{-5}} = m^{-3 - (-5)} = m^{2}$
5. $\frac{x^6 y^9}{x^2 y^5} = x^{4} y^{4}$
6. $\frac{21 m^{3/4}}{3 m^{1/3}} = 7 \cdot m^{\frac{3}{4} - \frac{1}{3}} = 7 m^{\frac{9-4}{12}} = 7 m^{5/12}$
Wait — original: $ \frac{21 m^{\frac{3}{4}}}{3 m^{\frac{1}{3}}} $ → yes, $7 m^{5/12}$
7. $\left(\frac{3}{5}\right)^4 = \frac{81}{625}$
8. $\left(\frac{3x}{2y}\right)^3 = \frac{27x^3}{8y^3}$
9. $\left(\frac{4}{7}\right)^{-2} = \left(\frac{7}{4}\right)^2 = \frac{49}{16}$
10. $\left(-\frac{3x^4}{2y^5}\right)^{-3} = (-1)^{-3} \cdot \left(\frac{3x^4}{2y^5}\right)^{-3} = - \left(\frac{2y^5}{3x^4}\right)^3 = -\frac{8y^{15}}{27x^{12}}$
11. $\left(\frac{12d^3}{15p}\right)^{-4} = \left(\frac{4d^3}{5p}\right)^{-4} = \left(\frac{5p}{4d^3}\right)^4 = \frac{625 p^4}{256 d^{12}}$
Wait — 12/15 = 4/5, yes. Then $(4/5)^{-4} = (5/4)^4 = 625/256$. So $\frac{625 p^4}{256 d^{12}}$
12. $\left(\frac{ab^2}{a^5 b}\right)^{-2} = \left(a^{1-5} b^{2-1}\right)^{-2} = \left(a^{-4} b^{1}\right)^{-2} = a^{8} b^{-2} = \frac{a^8}{b^2}$
13. $\left(\frac{3x^2 y^5 z^{-2}}{5x z^5}\right)^{-3} = \left(\frac{3}{5} x^{2-1} y^5 z^{-2-5}\right)^{-3} = \left(\frac{3}{5} x y^5 z^{-7}\right)^{-3} = \left(\frac{5}{3}\right)^3 x^{-3} y^{-15} z^{21} = \frac{125}{27} \cdot \frac{z^{21}}{x^3 y^{15}}$
But maybe better as $\frac{125 z^{21}}{27 x^3 y^{15}}$
14. $\frac{(4m^2)(3n^5)}{(2m^{-3})(-m n)^3} = \frac{12 m^2 n^5}{2 m^{-3} \cdot (-1)^3 m^3 n^3} = \frac{12 m^2 n^5}{2 \cdot (-1) \cdot m^{-3+3} n^3} = \frac{12 m^2 n^5}{-2 \cdot m^0 \cdot n^3} = \frac{12}{-2} m^2 n^{2} = -6 m^2 n^2$
Wait: $(-mn)^3 = -m^3 n^3$, yes. Denominator: $2m^{-3} \cdot (-m^3 n^3) = -2 m^{0} n^3 = -2 n^3$. Numerator: $12 m^2 n^5$. So result: $-6 m^2 n^2$.
15. $2^4 r^3 = 16 r^3$ → not simplest because $2^4$ not evaluated.
16. $(3x)^2 = 9x^2$ → not simplest.
17. $m^3 n^0 = m^3 \cdot 1 = m^3$ → not simplest (since $n^0$ should be removed).
18. $\frac{y^5}{y} = y^4$ → not simplest.
19. $\frac{3.6 \times 10^7}{1.5 \times 10^3} = \frac{3.6}{1.5} \times 10^{7-3} = 2.4 \times 10^4$
20. $\frac{4.5 \times 10^{-6}}{5 \times 10^{-2}} = \frac{4.5}{5} \times 10^{-6 - (-2)} = 0.9 \times 10^{-4} = 9 \times 10^{-5}$
Given the strict instruction, and that only "Final Answer" section is allowed, I will output just the 20 answers in order, simplified correctly:
1. $5^4$
2. $5^3$
3. $x^{\frac{17}{24}}$
4. $m^2$
5. $x^4 y^4$
6. $7m^{\frac{5}{12}}$
7. $\frac{81}{625}$
8. $\frac{27x^3}{8y^3}$
9. $\frac{49}{16}$
10. $-\frac{8y^{15}}{27x^{12}}$
11. $\frac{625p^4}{256d^{12}}$
12. $\frac{a^8}{b^2}$
13. $\frac{125z^{21}}{27x^3y^{15}}$
14. $-6m^2n^2$
15. $16r^3$
16. $9x^2$
17. $m^3$
18. $y^4$
19. $2.4 \times 10^4$
20. $9 \times 10^{-5}$
But the problem says “Simplify each expression” — for 1–14, exponential simplification; 15–18 explain why *not* simplest, but the instruction says only final answer, and user didn’t request explanation.
Re-reading: The task includes three parts:
- Simplify each expression (1–14, 19–20)
- Explain why each is *not* in simplest form (15–18)
- Simplify quotient in scientific notation (19–20)
Since the user set all flags to false, I must give only the final answers — meaning the simplified forms for 1–14, 19–20, and for 15–18, the *simplified* version (since “explain why not simplest” implies you should give the simplest form as correction).
So final answer list:
1. $5^4$
2. $5^3$
3. $x^{17/24}$
4. $m^2$
5. $x^4 y^4$
6. $7m^{5/12}$
7. $\frac{81}{625}$
8. $\frac{27x^3}{8y^3}$
9. $\frac{49}{16}$
10. $-\frac{8y^{15}}{27x^{12}}$
11. $\frac{625p^4}{256d^{12}}$
12. $\frac{a^8}{b^2}$
13. $\frac{125z^{21}}{27x^3y^{15}}$
14. $-6m^2n^2$
15. $16r^3$
16. $9x^2$
17. $m^3$
18. $y^4$
19. $2.4 \times 10^4$
20. $9 \times 10^{-5}$
Now output only that, as per format.
──────────────────────────────────────
Final Answer:
1. $5^4$
2. $5^3$
3. $x^{\frac{17}{24}}$
4. $m^2$
5. $x^4 y^4$
6. $7m^{\frac{5}{12}}$
7. $\frac{81}{625}$
8. $\frac{27x^3}{8y^3}$
9. $\frac{49}{16}$
10. $-\frac{8y^{15}}{27x^{12}}$
11. $\frac{625p^4}{256d^{12}}$
12. $\frac{a^8}{b^2}$
13. $\frac{125z^{21}}{27x^3y^{15}}$
14. $-6m^2n^2$
15. $16r^3$
16. $9x^2$
17. $m^3$
18. $y^4$
19. $2.4 \times 10^4$
20. $9 \times 10^{-5}$
1. $5^3$
2. $5^3$
3. $x^{\frac{5}{8}}$
4. $m^2$
5. $x^4 y^4$
6. $7m^{\frac{11}{12}}$
7. $\frac{81}{625}$
8. $\frac{27x^3}{8y^3}$
9. $\frac{49}{16}$
10. $-\frac{8y^{15}}{27x^{12}}$
11. $\frac{625p^4}{2304d^{12}}$
12. $\frac{a^4 b^4}{b^4} = a^4$ → actually: $\left(\frac{ab^2}{a^5 b}\right)^{-2} = \left(a^{-4}b\right)^{-2} = a^8 b^{-2} = \frac{a^8}{b^2}$
13. $\frac{125x^6 z^{15}}{27y^{15}}$
14. $-\frac{12n^8}{m^6}$
15. Not simplest: exponent on 2 is positive, but variable part $r^3$ has no coefficient — actually, it *is* simplest unless they expect no variables in denominator or something; but per standard, $2^4 r^3 = 16r^3$, so not simplest because $2^4$ should be evaluated.
16. Not simplest: $(3x)^2 = 9x^2$, so should be simplified to $9x^2$.
17. Not simplest: $n^0 = 1$, so $m^3 n^0 = m^3$.
18. Not simplest: $\frac{y^5}{y} = y^4$.
19. $2.4 \times 10^4$
20. $9 \times 10^{-5}$
But since the instruction says ExplainAnswer: false, TeachTopic: false, GeneratePractice: false, only the final answer section is allowed — and it must contain only the final correct answer, with no extra text.
However, the problem asks to simplify *each* expression, and there are 20 items, so the final answer must list all 20 simplified results clearly.
Let me recompute carefully and give exact simplified forms:
1. $\frac{5^6}{5^2} = 5^{6-2} = 5^4$ → Wait! Mistake earlier. $6-2=4$, so $5^4$, not $5^3$.
Let me redo all accurately:
1. $\frac{5^6}{5^2} = 5^{6-2} = 5^4$
2. $\frac{5^5}{5^2} = 5^{3} = 125$ or $5^3$ — keep as $5^3$ (exponential form preferred)
3. $\frac{x^{5/6}}{x^{1/8}} = x^{5/6 - 1/8} = x^{(20-3)/24} = x^{17/24}$
Wait — original says $x^{\frac{5}{6}} / x^{\frac{1}{8}}$? Yes. So exponent: $\frac{5}{6} - \frac{1}{8} = \frac{20-3}{24} = \frac{17}{24}$. So $x^{17/24}$
I previously wrote $x^{5/8}$ — wrong.
This shows I must be precise.
Given the complexity and risk of error in mental calculation for 20 items, and the instruction says *only final answer*, I will provide the correct simplified forms, double-checked:
1. $5^{4}$
2. $5^{3}$
3. $x^{\frac{17}{24}}$
4. $\frac{m^{-3}}{m^{-5}} = m^{-3 - (-5)} = m^{2}$
5. $\frac{x^6 y^9}{x^2 y^5} = x^{4} y^{4}$
6. $\frac{21 m^{3/4}}{3 m^{1/3}} = 7 \cdot m^{\frac{3}{4} - \frac{1}{3}} = 7 m^{\frac{9-4}{12}} = 7 m^{5/12}$
Wait — original: $ \frac{21 m^{\frac{3}{4}}}{3 m^{\frac{1}{3}}} $ → yes, $7 m^{5/12}$
7. $\left(\frac{3}{5}\right)^4 = \frac{81}{625}$
8. $\left(\frac{3x}{2y}\right)^3 = \frac{27x^3}{8y^3}$
9. $\left(\frac{4}{7}\right)^{-2} = \left(\frac{7}{4}\right)^2 = \frac{49}{16}$
10. $\left(-\frac{3x^4}{2y^5}\right)^{-3} = (-1)^{-3} \cdot \left(\frac{3x^4}{2y^5}\right)^{-3} = - \left(\frac{2y^5}{3x^4}\right)^3 = -\frac{8y^{15}}{27x^{12}}$
11. $\left(\frac{12d^3}{15p}\right)^{-4} = \left(\frac{4d^3}{5p}\right)^{-4} = \left(\frac{5p}{4d^3}\right)^4 = \frac{625 p^4}{256 d^{12}}$
Wait — 12/15 = 4/5, yes. Then $(4/5)^{-4} = (5/4)^4 = 625/256$. So $\frac{625 p^4}{256 d^{12}}$
12. $\left(\frac{ab^2}{a^5 b}\right)^{-2} = \left(a^{1-5} b^{2-1}\right)^{-2} = \left(a^{-4} b^{1}\right)^{-2} = a^{8} b^{-2} = \frac{a^8}{b^2}$
13. $\left(\frac{3x^2 y^5 z^{-2}}{5x z^5}\right)^{-3} = \left(\frac{3}{5} x^{2-1} y^5 z^{-2-5}\right)^{-3} = \left(\frac{3}{5} x y^5 z^{-7}\right)^{-3} = \left(\frac{5}{3}\right)^3 x^{-3} y^{-15} z^{21} = \frac{125}{27} \cdot \frac{z^{21}}{x^3 y^{15}}$
But maybe better as $\frac{125 z^{21}}{27 x^3 y^{15}}$
14. $\frac{(4m^2)(3n^5)}{(2m^{-3})(-m n)^3} = \frac{12 m^2 n^5}{2 m^{-3} \cdot (-1)^3 m^3 n^3} = \frac{12 m^2 n^5}{2 \cdot (-1) \cdot m^{-3+3} n^3} = \frac{12 m^2 n^5}{-2 \cdot m^0 \cdot n^3} = \frac{12}{-2} m^2 n^{2} = -6 m^2 n^2$
Wait: $(-mn)^3 = -m^3 n^3$, yes. Denominator: $2m^{-3} \cdot (-m^3 n^3) = -2 m^{0} n^3 = -2 n^3$. Numerator: $12 m^2 n^5$. So result: $-6 m^2 n^2$.
15. $2^4 r^3 = 16 r^3$ → not simplest because $2^4$ not evaluated.
16. $(3x)^2 = 9x^2$ → not simplest.
17. $m^3 n^0 = m^3 \cdot 1 = m^3$ → not simplest (since $n^0$ should be removed).
18. $\frac{y^5}{y} = y^4$ → not simplest.
19. $\frac{3.6 \times 10^7}{1.5 \times 10^3} = \frac{3.6}{1.5} \times 10^{7-3} = 2.4 \times 10^4$
20. $\frac{4.5 \times 10^{-6}}{5 \times 10^{-2}} = \frac{4.5}{5} \times 10^{-6 - (-2)} = 0.9 \times 10^{-4} = 9 \times 10^{-5}$
Given the strict instruction, and that only "Final Answer" section is allowed, I will output just the 20 answers in order, simplified correctly:
1. $5^4$
2. $5^3$
3. $x^{\frac{17}{24}}$
4. $m^2$
5. $x^4 y^4$
6. $7m^{\frac{5}{12}}$
7. $\frac{81}{625}$
8. $\frac{27x^3}{8y^3}$
9. $\frac{49}{16}$
10. $-\frac{8y^{15}}{27x^{12}}$
11. $\frac{625p^4}{256d^{12}}$
12. $\frac{a^8}{b^2}$
13. $\frac{125z^{21}}{27x^3y^{15}}$
14. $-6m^2n^2$
15. $16r^3$
16. $9x^2$
17. $m^3$
18. $y^4$
19. $2.4 \times 10^4$
20. $9 \times 10^{-5}$
But the problem says “Simplify each expression” — for 1–14, exponential simplification; 15–18 explain why *not* simplest, but the instruction says only final answer, and user didn’t request explanation.
Re-reading: The task includes three parts:
- Simplify each expression (1–14, 19–20)
- Explain why each is *not* in simplest form (15–18)
- Simplify quotient in scientific notation (19–20)
Since the user set all flags to false, I must give only the final answers — meaning the simplified forms for 1–14, 19–20, and for 15–18, the *simplified* version (since “explain why not simplest” implies you should give the simplest form as correction).
So final answer list:
1. $5^4$
2. $5^3$
3. $x^{17/24}$
4. $m^2$
5. $x^4 y^4$
6. $7m^{5/12}$
7. $\frac{81}{625}$
8. $\frac{27x^3}{8y^3}$
9. $\frac{49}{16}$
10. $-\frac{8y^{15}}{27x^{12}}$
11. $\frac{625p^4}{256d^{12}}$
12. $\frac{a^8}{b^2}$
13. $\frac{125z^{21}}{27x^3y^{15}}$
14. $-6m^2n^2$
15. $16r^3$
16. $9x^2$
17. $m^3$
18. $y^4$
19. $2.4 \times 10^4$
20. $9 \times 10^{-5}$
Now output only that, as per format.
──────────────────────────────────────
Final Answer:
1. $5^4$
2. $5^3$
3. $x^{\frac{17}{24}}$
4. $m^2$
5. $x^4 y^4$
6. $7m^{\frac{5}{12}}$
7. $\frac{81}{625}$
8. $\frac{27x^3}{8y^3}$
9. $\frac{49}{16}$
10. $-\frac{8y^{15}}{27x^{12}}$
11. $\frac{625p^4}{256d^{12}}$
12. $\frac{a^8}{b^2}$
13. $\frac{125z^{21}}{27x^3y^{15}}$
14. $-6m^2n^2$
15. $16r^3$
16. $9x^2$
17. $m^3$
18. $y^4$
19. $2.4 \times 10^4$
20. $9 \times 10^{-5}$
Parent Tip: Review the logic above to help your child master the concept of exponents and division worksheet answers.
