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1.
a) \(3^4\times3^5\times3^6=3^{4+5+6}=3^{15}\)
b) \(5^2\times5^4\times5^5\times25=5^2\times5^4\times5^5\times5^2=5^{2+4+5+2}=5^{13}\)
c) \(10^8\div10^3=10^{8-3}=10^5\)
d) \(a^7\div a^2=a^{7-2}=a^5\)
2.
\(987=900+80+7\\ =9\times100+8\times10+7\\ =9\times10^2+8\times10^1+7\times10^0\)
\(2021=2000+20+1\\ =2\times1000+2\times10+1\times1\\ =2\times10^3+2\times10^1+1\times10^0\)
\(abcde=a\times10000+b\times1000+c\times100+d\times10+e\times1\\ =a\times10^4+b\times10^3+c\times10^2+d\times10^1+e\times10^0\)
1) a) 4⁸.2²⁰ = (2²)⁸.2²⁰
= 2¹⁶.2²⁰ = 2³⁶
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9¹².27⁵.81³ = (3²)¹².(3³)⁵.(3⁴)⁴
= 3²⁴.3¹⁵.3¹⁶ = 3⁵⁵
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64³.4⁵.16² = (4³)³.4⁵.(4²)²
= 4⁹.4⁵.4⁴ = 4¹⁸
b) 25²⁰.125⁴ = (5²)²⁰.(5³)⁴
= 5⁴⁰.5¹² = 5⁵²
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x⁷.x³.x⁴ = x¹⁴
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3⁶.4⁶ = (3.4)⁶ = 12⁶
2) a) 2² = 4
2³ = 8
2⁴ = 16
2⁵ = 32
2⁶ = 64
2⁷ = 128
2⁸ = 256
2⁹ = 512
2¹⁰ = 1024
b) 3² = 9
3³ = 27
3⁴ = 81
3⁵ = 243
c) 4² = 16
4³ = 64
4⁴ = 256
d) 5² = 25
5³ = 125
5⁴ = 625
1: 2*2*2*2*2*2=2^6
2: x*x*x*x=x^4
3: =7^4*6^4=42^4
4: =5^2*3^2*4^2=60^2
5: =2*3^2*5^5
6: =10^3*10^3=10^6
a) 23 x 24 x 2 = 23+4+1 = 28
b) 52 x 57 x 53 = 52+7+3 = 512
c) 62 x 6 x 63 x 62 = 62+1+3+2 = 68
d) a3 x a5 x a7 x a = a3+5+7+1 = a16
a: \(3\cdot3\cdot3\cdot3\cdot3=3^5\)
b: \(y\cdot y\cdot y\cdot y=y^4\)
c: \(5\cdot p\cdot5\cdot p\cdot2\cdot q\cdot4\cdot q=25\cdot2\cdot4\cdot p^2q^2=2\cdot\left(10qp\right)^2\)
d: \(a\cdot a+b\cdot b+c\cdot c+d\cdot d\cdot d\cdot d=a^2+b^2+c^2+d^4\)
25.84=25.(23)4=25.212=217
256.1253=(52)6.(53)3=512.59=517
6255:257=(54)5:(52)7=520:214=56
1) \(9\cdot9\cdot9\cdot9\cdot3\cdot3\)
\(=3\cdot3\cdot3\cdot3\cdot3\cdot3\cdot3\cdot3\cdot3\cdot3\)
\(=3^{10}\)
2) \(8\cdot2\cdot2\cdot2\cdot16\cdot4\)
\(=2\cdot2\cdot2\cdot2\cdot2\cdot2\cdot2\cdot2\cdot2\cdot2\cdot2\cdot2\)
\(=2^{12}\)
3) \(3\cdot3\cdot3\cdot3\cdot5\cdot5\cdot5\cdot5\)
\(=3^4\cdot5^4\)
4) \(25\cdot25\cdot25\cdot5\)
\(=5\cdot5\cdot5\cdot5\cdot5\cdot5\cdot5\)
\(=5^7\)
5) \(7\cdot49\cdot7\cdot7\cdot7\)
\(=7\cdot7\cdot7\cdot7\cdot7\cdot7\)
\(=7^6\)
6) \(8\cdot64\cdot8\cdot64\)
\(=2\cdot2\cdot2\cdot2\cdot2\cdot2\cdot2\cdot2\cdot2\cdot2\cdot2\cdot2\cdot2\cdot2\cdot2\cdot2\cdot2\cdot2\)
\(=2^{18}\)
7) \(2\cdot2\cdot2\cdot2\cdot2\cdot2\cdot4\)
\(=2\cdot2\cdot2\cdot2\cdot2\cdot2\cdot2\cdot2\)
\(=2^8\)
1) 310
2) 212
3)34 . 54
4) 57
5) 76
6)86
7) 28.