Question

write the following without using exponents and then simplify. power

1. $(3^{3})^{4}$
2. $(3^{2})^{3}$
3. $(4^{2})^{2}$
4. $(2^{3})^{3}$
5. $(x^{4})^{2}$
6. $(4b^{2})^{3}$
7. $(4n)^{4}$
8. $(4v^{4})^{3}$
9. $(2m^{2}n^{3})^{4}$
10. $(4xy)^{2}$

write the following without using exponents and then simplify. quotient (divide)

1. $\frac{4^{4}}{4^{2}}$
2. $\frac{4^{5}}{4^{3}}$
3. $\frac{6^{6}}{6^{2}}$
4. $\frac{4^{6}}{4}$
5. $\frac{2k^{6}}{4k^{2}}$
6. $\frac{x^{5}}{x^{2}}$
7. $\frac{4x^{6}}{4x^{4}}$
8. $\frac{6p^{5}}{4p^{2}}$
9. $\frac{2u^{4}v^{3}}{u^{3}}$
10. $\frac{4u^{3}v^{7}}{u^{2}v^{4}}$

Explanation:

Step1: 幂的乘方：指数相乘

对于幂的乘方形式 \((a^m)^n\)，根据规则转化为 \(a^{m \times n}\)，再展开计算；对于积的乘方 \((ab)^n\)，转化为 \(a^n b^n\) 后展开。

23) \((3^5)^4\)

\(3^{5 \times 4}=3^{20}=3\times3\times3\times\cdots\times3\)（共20个3相乘）\(=3486784401\)

24) \((3^2)^3\)

\(3^{2 \times 3}=3^6=3\times3\times3\times3\times3\times3=729\)

25) \((4^2)^2\)

\(4^{2 \times 2}=4^4=4\times4\times4\times4=256\)

26) \((2^3)^3\)

\(2^{3 \times 3}=2^9=2\times2\times\cdots\times2\)（共9个2相乘）\(=512\)

27) \((x^4)^2\)

\(x^{4 \times 2}=x^8=x\times x\times\cdots\times x\)（共8个x相乘）

28) \((4b^2)^3\)

\(4^3 \times (b^2)^3=64 \times b^{6}=64b\times b\times b\times b\times b\times b\)

29) \((4n)^4\)

\(4^4 \times n^4=256 \times n\times n\times n\times n=256n^4\)

30) \((4v^4)^3\)

\(4^3 \times (v^4)^3=64 \times v^{12}=64v\times v\times\cdots\times v\)（共12个v相乘）

31) \((2m^2n^3)^4\)

\(2^4 \times (m^2)^4 \times (n^3)^4=16m^8n^{12}=16\times m^8\times n^{12}\)

32) \((4xy)^2\)

\(4^2 \times x^2 \times y^2=16x^2y^2=16\times x\times x\times y\times y\)

Step2: 同底数幂除法：指数相减

对于同底数幂除法 \(\frac{a^m}{a^n}\)（\(a
eq0\)），转化为 \(a^{m-n}\)，再展开或化简系数。

33) \(\frac{4^4}{4^2}\)

\(4^{4-2}=4^2=4\times4=16\)

34) \(\frac{4^5}{4^3}\)

\(4^{5-3}=4^2=4\times4=16\)

35) \(\frac{6^6}{6^2}\)

\(6^{6-2}=6^4=6\times6\times6\times6=1296\)

36) \(\frac{4^6}{4}\)

\(4^{6-1}=4^5=4\times4\times4\times4\times4=1024\)

37) \(\frac{2k^6}{4k^2}\)

\(\frac{2}{4} \times k^{6-2}=\frac{1}{2}k^4=\frac{1}{2}\times k\times k\times k\times k\)

38) \(\frac{x^5}{x^2}\)

\(x^{5-2}=x^3=x\times x\times x\)

39) \(\frac{4x^6}{4x^4}\)

\(\frac{4}{4} \times x^{6-4}=x^2=x\times x\)

40) \(\frac{6p^5}{4p^2}\)

\(\frac{6}{4} \times p^{5-2}=\frac{3}{2}p^3=\frac{3}{2}\times p\times p\times p\)

41) \(\frac{2u^4v^3}{u^3}\)

\(2 \times u^{4-3} \times v^3=2uv^3=2\times u\times v\times v\times v\)

42) \(\frac{4u^3v^7}{u^2v^4}\)

\(4 \times u^{3-2} \times v^{7-4}=4uv^3=4\times u\times v\times v\times v\)

Answer:

1. \(3486784401\)
2. \(729\)
3. \(256\)
4. \(512\)
5. \(x^8\)
6. \(64b^6\)
7. \(256n^4\)
8. \(64v^{12}\)
9. \(16m^8n^{12}\)
10. \(16x^2y^2\)
11. \(16\)
12. \(16\)
13. \(1296\)
14. \(1024\)
15. \(\frac{1}{2}k^4\)
16. \(x^3\)
17. \(x^2\)
18. \(\frac{3}{2}p^3\)
19. \(2uv^3\)
20. \(4uv^3\)