Question

directions: simplify the following monomials. 1. 7k + 4k 2. 11x^4 - 3x^4 3. -2m^2n^2 - 3m^2n^2 4. 3p^3+(-4p^3) 5. -14c^7d^2 + 7c^7d^2 6. -10rs-(-14rs) 7. -8a - b - 2b + 3a 8. 6x - 4x^2 - x - 7 + 5x^2 9. 10cd + 2c^2 - d^2 + 4d^2 - 11cd 10. find the difference of 9p^3q^2 and 13p^3q^2. 11. subtract -2x^2 from the sum of 4x^2 and 15x^2. directions: simplify the following monomials. express final answers using only positive exponents. 12. 2^9·2^13 13. 8·8^-5 14. (-10)^6·(-10)^-6 15. a^4·a^2 16. k^3·k^-8 17. x^8·x^-1 18. w^5·w^-7·w 19. c^-5·d^-1·c^4 20. p^-2·q^-1·p^11·q^-6 21. 5x^4·7x^3 22. (9k^5)·(-2k^-2) 23. -4v^3·-3v^-10 24. (2xy)·(5x^8y^3) 25. 2/3 a^-2b^-1·-18a^-3b^1 26. (-4m^5n^7)(mn)(-2m^2n^6) 27. abc^3·a^6b^4 + 9a^7b^5c^3 28. 10h^10·2h^-1 - 3h^5·h^4

Explanation:

Step1: Combine like - terms for addition/subtraction of monomials

1. \(7k + 4k=(7 + 4)k=11k\)
2. \(11x^{4}-3x^{4}=(11 - 3)x^{4}=8x^{4}\)
3. \(-2m^{2}n^{3}-3m^{2}n^{3}=(-2-3)m^{2}n^{3}=-5m^{2}n^{3}\)
4. \(3p^{3}+(-4p^{3})=(3 - 4)p^{3}=-p^{3}\)
5. \(-14c^{7}d^{2}+7c^{7}d^{2}=(-14 + 7)c^{7}d^{2}=-7c^{7}d^{2}\)
6. \(-10rs-(-14rs)=-10rs + 14rs=(-10 + 14)rs = 4rs\)
7. \(-8a - b-2b+3a=(-8a+3a)+(-b - 2b)=-5a-3b\)
8. \(6x-4x^{2}-x - 7+5x^{2}=(-4x^{2}+5x^{2})+(6x - x)-7=x^{2}+5x - 7\)
9. \(10cd+2c^{2}-d^{2}+4d^{2}-11cd=2c^{2}+(-d^{2}+4d^{2})+(10cd-11cd)=2c^{2}+3d^{2}-cd\)
10. \(9p^{3}q^{2}-13p^{3}q^{2}=(9 - 13)p^{3}q^{2}=-4p^{3}q^{2}\)
11. \((4x^{2}+15x^{2})-(-2x^{2})=(4 + 15)x^{2}+2x^{2}=19x^{2}+2x^{2}=21x^{2}\)

Step2: Use the rule \(a^{m}\cdot a^{n}=a^{m + n}\) for multiplication of monomials and simplify with positive exponents

1. \(2^{9}\cdot2^{13}=2^{9 + 13}=2^{22}\)
2. \(8\cdot8^{-5}=8^{1+( - 5)}=8^{-4}=\frac{1}{8^{4}}=\frac{1}{4096}\)
3. \((-10)^{6}\cdot(-10)^{-6}=(-10)^{6+( - 6)}=(-10)^{0}=1\)
4. \(a^{4}\cdot a^{2}=a^{4 + 2}=a^{6}\)
5. \(k^{3}\cdot k^{-8}=k^{3+( - 8)}=k^{-5}=\frac{1}{k^{5}}\)
6. \(x^{8}\cdot x^{-1}=x^{8+( - 1)}=x^{7}\)
7. \(w^{5}\cdot w^{-7}\cdot w=w^{5+( - 7)+1}=w^{-1}=\frac{1}{w}\)
8. \(c^{-5}\cdot d^{-1}\cdot c^{4}=c^{-5 + 4}\cdot d^{-1}=c^{-1}d^{-1}=\frac{1}{cd}\)
9. \(p^{-2}\cdot q^{-1}\cdot p^{11}\cdot q^{-6}=p^{-2 + 11}\cdot q^{-1+( - 6)}=p^{9}q^{-7}=\frac{p^{9}}{q^{7}}\)
10. \(5x^{4}\cdot7x^{3}=(5\times7)x^{4 + 3}=35x^{7}\)
11. \((9k^{5})\cdot(-2k^{-2})=9\times(-2)k^{5+( - 2)}=-18k^{3}\)
12. \(-4v^{3}\cdot(-3v^{-10})=(-4)\times(-3)v^{3+( - 10)} = 12v^{-7}=\frac{12}{v^{7}}\)
13. \((2xy)\cdot(5x^{8}y^{3})=(2\times5)x^{1 + 8}y^{1+3}=10x^{9}y^{4}\)
14. \(\frac{2}{3}a^{-2}b^{-1}\cdot(-18a^{-3}b^{1})=\frac{2}{3}\times(-18)a^{-2+( - 3)}b^{-1 + 1}=-12a^{-5}=\frac{-12}{a^{5}}\)
15. \((-4m^{5}n^{7})(mn)(-2m^{2}n^{6})=(-4)\times(-2)m^{5 + 1+2}n^{7+1 + 6}=8m^{8}n^{14}\)
16. \(abc^{3}\cdot a^{6}b^{4}+9a^{7}b^{5}c^{3}=a^{1+6}b^{1 + 4}c^{3}+9a^{7}b^{5}c^{3}=a^{7}b^{5}c^{3}+9a^{7}b^{5}c^{3}=10a^{7}b^{5}c^{3}\)
17. \(10h^{10}\cdot2h^{-1}-3h^{5}\cdot h^{4}=10\times2h^{10+( - 1)}-3h^{5 + 4}=20h^{9}-3h^{9}=17h^{9}\)

Answer:

1. \(11k\)
2. \(8x^{4}\)
3. \(-5m^{2}n^{3}\)
4. \(-p^{3}\)
5. \(-7c^{7}d^{2}\)
6. \(4rs\)
7. \(-5a-3b\)
8. \(x^{2}+5x - 7\)
9. \(2c^{2}+3d^{2}-cd\)
10. \(-4p^{3}q^{2}\)
11. \(21x^{2}\)
12. \(2^{22}\)
13. \(\frac{1}{4096}\)
14. \(1\)
15. \(a^{6}\)
16. \(\frac{1}{k^{5}}\)
17. \(x^{7}\)
18. \(\frac{1}{w}\)
19. \(\frac{1}{cd}\)
20. \(\frac{p^{9}}{q^{7}}\)
21. \(35x^{7}\)
22. \(-18k^{3}\)
23. \(\frac{12}{v^{7}}\)
24. \(10x^{9}y^{4}\)
25. \(\frac{-12}{a^{5}}\)
26. \(8m^{8}n^{14}\)
27. \(10a^{7}b^{5}c^{3}\)
28. \(17h^{9}\)