Exercise 1

Calculate:

1   3^3 \cdot 3^4 \cdot 3

2   5^7 \cdot 5^3

3   (5^3)^4

4   (5 \cdot 2 \cdot 3)^4

(3^4)^4

[(5^3)^4]^2

(8^2)^3

(9^3)^2

2^5 \cdot 2 ^4 \cdot 2

10  2^7 : 2 ^6

11  (2^2)^4

12  (4 \cdot 2 \cdot 3)^4 = 2 (-8_ \cdot (-2)^2 \cdot (2)^0 (-2)

13  (2^5)^4 = 1(-2)^2 \cdot (-2)^3 \cdot (-2)^4

14  [(2^3)^4]^0

15  (27^2)^5

16  (4^3)^2

Exercise 2

Calculate:

1.  (-2)^2 \cdot (-2)^3 \cdot (-2)^4

2 ^ {-2}\cdot 2 ^ {-3} \cdot 2 ^4

2 ^2 : 2^3

2 ^ {-2} : 2^3

2^2 : 2^ {-3}

2^ {-2} : 2 ^ {-3} = 2

[(-2)^ {-2}]^3 \cdot (-2)^3 \cdot (-2)^4

[(-2)^6 : (-2)^3]^3 \cdot (-2) \cdot (-2) ^ {-4}

Exercise 3

Calculate:

(-3)^1 \cdot (-3)^3 \cdot (-3)^4

(-27) \cdot (-3) \cdt (-3) ^2 \cdot (-3)^0

(-3)^2 \cdot (-3)^3 \cdot (-3)^ {-4}

3 ^ {-2} \cdot 3 ^ {-4} \cdot 3^4

5   5 ^ {-2} : 5 ^3

6   5^2 : 5^ {-3}

5^ {-2} : 5 ^ {-3}

(-3)^1 \cdot [(-3)^3]^2 \cdot (-3)^ {-4}

Exercise 4

Calculate:

(\frac{2}{3})^2 \cdot (\frac{2}{3})^3

(\frac{2}{3})^{-2} \cdot (\frac{2}{3})^3

(\frac{2}{3})^2 \cdot (\frac{2}{3})^{-3}

(\frac{2}{3})^{-2} \cdot (\frac{2}{3})^{-3}

(\frac{2}{3})^{-2} \cdot (\frac{3}{2})^{-3}

(\frac{2}{3})^2 : (\frac{2}{3})^3

(\frac{2}{3})^{-2} : (\frac{2}{3})^3

(\frac{2}{3})^2 : (\frac{2}{3})^{-3}

(\frac{2}{3})^{-2} : (\frac{2}{3})^{-3}

10  (\frac{3}{2})^{-2}: (\frac{2}{3})^{-3}

11 [(\frac{2}{3})^2]^3

12  {[(\frac{2}{3})^2]^3}^{-4}

13  (\frac{4}{9})^{-2} : (\frac{27}{8})^ {-3}

Exercise 5

Calculate:

\frac{(\frac{2}{3})^5 (\frac{2}{3})^0 (\frac{2}{3})^{-3} (\frac{81}{16})^ {-2}} {(\frac{3}{2})^{-5} (\frac{2}{3}) [(\frac{2}{3})^5]^2 (\frac{8}{27})^3}

Exercise 6

Calculate:

\frac{(2 - \frac{1}{5})^2} {(3 - \frac{2}{9})^{-1}} : \frac{(\frac{6}{7} \cdot \frac{5}{4} - \frac{2}{7} : \frac{1}{2})^3} { (\frac{1}{2} - \frac{1}{3} \cdot \frac{1}{4} : \frac{1}{5})} - 5 \frac{1}{7}

Exercise 7

Calculate:

1   16 ^ {\frac{3}{2}}

8 ^ {\frac{2}{3}}

81 ^ {0.75}

8 ^ {0.333...}

 

Solution of exercise 1

Calculate:

3^3 \cdot 3 ^4 \cdot 3 = 3 ^8

5^7 : 5^3 = 5^4

3   (5^3)^4 = 5 ^{12}

(5 \cdot 2 \cdot 3)^4 = 30^4

5(3^4)^4 = 3^ {16}

[(5^3)^4]^2 = (5 ^ {12})^2 = 5^ {24}

(8^2)^3 = [(2^3)^2]^3 = (2^6)^3 = 2^{18}

(9^3)^2 = [ (3^2)^3]^2 = (3^6)^2 = 3^{12}

2^5 \cdot 2^4 \cdot 2 = 2 ^ {10}

10  2^7 : 2 ^6 = 2

11  (2^2)^4 = 2^8

12  (4 \cdot 2 \cdot 3)^4 = 24^4

13  (2^5)^4 = 2^ {20}

14 [(2^3)^4]^0 = (2^ {12})^0 = 2^0 = 1

15 (27)^2)^5 = [(3^3)^2]^5 = (3^6)^5 = 3^{30}

16 (4^3)^2 = [(2^2)^3]^2 = (2^6)^2 = 2^{12}

 

Solution of exercise 2

1.  (-2)^2 \cdot (-2)^3 \cdot (-2)^4 = (-2)^9 = -512

2 ^ {-2} \cdot 2 ^ {-3} \cdot 2 ^4 = 2 ^ {-1} = \frac{1}{2}

2 ^2 : 2^3 = 2 ^ {-1} = \frac{1}{2}

2 ^ {-2} : 2^3 = 2 ^ {-5} = \frac{1}{2^5}

2^2 : 2^ {-3} = 2^5

2^ {-2} : 2 ^ {-3} = 2

[(-2)^ {-2}]^3 \cdot (-2)^3 \cdot (-2)^4 = \frac{1}{64} \cdot -8 \cdot 16 = -2

[(-2)^6 : (-2)^3]^3 \cdot (-2) \cdot (-2) ^ {-4} = 64

 

Solution of exercise 3

(-3)^1 \cdot (-3)^3 \cdot (-3)^4 = (-3)^8 = 6561

(-27) \cdot (-3) \cdt (-3) ^2 \cdot (-3)^0 = (-3)^6 = 729

(-3)^2 \cdot (-3)^3 \cdot (-3)^ {-4} = -3

3 ^ {-2} \cdot 3 ^ {-4} \cdot 3^4 = (3)^{-2} = \frac{1}{9}

5   5 ^ {-2} : 5 ^3 = 5^ {-5} = \frac{1}{3125}

6   5^2 : 5^ {-3} = 5^5 = 3125

5^ {-2} : 5 ^ {-3} = 5

(-3)^1 \cdot [(-3)^3]^2 \cdot (-3)^ {-4} = (-3) \cdot (-27)^2 \cdot \frac{1}{(-3)^{4}} = (-3) \cdot 729 \cdot \frac{1}{81} = -27

 

Solution of exercise 4

Calculate:

(\frac{2}{3})^2 \cdot (\frac{2}{3})^3 = (\frac{2}{3})^5

2.  (\frac{2}{3})^{-2} \cdot (\frac{2}{3})^3 = \frac{2}{3}

3  3  (\frac{2}{3})^2 \cdot (\frac{2}{3})^{-3} = (\frac{2}{3})^{-1} = \frac{3}{2}

(\frac{2}{3})^{-2} \cdot (\frac{2}{3})^{-3} = (\frac{2}{3})^{-5} = (\frac{3}{2})^5

(\frac{2}{3})^{-2} \cdot (\frac{3}{2})^{-3} = (\frac{2}{3})^ {-2} \cdot (\frac{2}{3})^3 = \frac{2}{3}

(\frac{2}{3})^2 : (\frac{2}{3})^3 = (\frac{2}{3})^{-1} = \frac{3}{2}

(\frac{2}{3})^{-2} : (\frac{2}{3})^3 = (\frac{2}{3})^{-5} = (\frac{3}{2})^5

(\frac{2}{3})^2 : (\frac{2}{3})^{-3} = (\frac{2}{3})^5

(\frac{2}{3})^{-2} : (\frac{2}{3})^{-3} = \frac{2}{3}

10  (\frac{3}{2})^{-2}: (\frac{2}{3})^{-3} = (\frac{2}{3})^2 : (\frac{2}{3})^{-3} = (\frac{2}{3})^5

11 [(\frac{2}{3})^2]^3 = (\frac{2}{3})^6

12  12  {[(\frac{2}{3})^2]^3}^{-4} = (\frac{2}{3})^{-24} = (\frac{3}{2})^{24}

13   (\frac{4}{9})^{-2} : (\frac{27}{8})^ {-3}

=[(\frac{2}{3})^2]^{-2} : [ (\frac{3}{2})^3]^{-3} = (\frac{2}{3})^{-4} : (\frac{3}{2})^ {-9} = (\frac{2}{3})^{-4} : (\frac{2}{3})^9 = (\frac{2}{3})^{-13} = (\frac{3}{2})^{13}

 

Solution of exercise 5

Calculate:

\frac{(\frac{2}{3})^5 (\frac{2}{3})^0 (\frac{2}{3})^{-3} (\frac{81}{16})^ {-2}} {(\frac{3}{2})^{-5} (\frac{2}{3}) [(\frac{2}{3})^5]^2 (\frac{8}{27})^3}

 

= \frac{(\frac{2}{3})^5 (\frac{2}{3})^0 (\frac{2}{3})^{-3} [(\frac{3}{2})^4] {-2}} {(\frac{3}{2})^{-5} (\frac{2}{3}) [(\frac{2}{3})^5]^2 [(\frac{2}{3})^3]^3}

 

= \frac{(\frac{2}{3})^5 (\frac{2}{3})^0 (\frac{2}{3})^{-3} (\frac{2}{3})^ {8}} {(\frac{3}{2})^{-5} (\frac{2}{3}) (\frac{2}{3})^{10} (\frac{2}{3})^9}

 

=  \frac{(\frac{2}{3})^5 (\frac{2}{3})^0 (\frac{2}{3})^{-3} (\frac{3}{2})^ {-8}} {(\frac{2}{3})^{5} (\frac{2}{3}) (\frac{2}{3})^{10} (\frac{2}{3})^9}

 

= \frac{(\frac{2}{3})^{10}} { (\frac{2}{3})^{25}} = (\frac{2}{3})^ {-15} = (\frac{3}{2})^{15}

 

Solution of exercise 6

Calculate:

\frac{(2 - \frac{1}{5})^2} {(3 - \frac{2}{9})^{-1}} : \frac{(\frac{6}{7} \cdot \frac{5}{4} - \frac{2}{7} : \frac{1}{2})^3} { (\frac{1}{2} - \frac{1}{3} \cdot \frac{1}{4} : \frac{1}{5})} - 5 \frac{1}{7}

 

= \frac{(\frac{10 - 1}{5})^2} {(\frac{27 - 2}{9})^{-1}} : \frac { (\frac{30}{28} - \frac{4}{7})^3} {(\frac{1}{2} - \frac{1}{12} : \frac{1}{5})} - \frac{35 + 1} {7}

 

= \frac{(\frac{9}{5})^2} {(\frac{25}{9})^{-1}} : \frac{( \frac{15}{4} - \frac{4}{7})^3} {(\frac{1}{2} - \frac{5}{12})} - \frac{36}{7}

= \frac{(\frac{9}{5})^2} {(\frac{25}{9})^{-1}} : \frac {(\frac{15 - 8}{14})^3} { (\frac{6 - 5}{12})} - \frac{36}{7}

 

= \frac{(\frac{9}{5})^2} {(\frac{25}{9})^{-1}} : \frac{ (\frac{1}{2})^3} { \frac{1}{12}} - \frac{36}{7}

 

= \frac{81}{25}\frac{9}{25} : \frac{1}{8}\frac{1}{12} - \frac{36}{7} = \frac{81}{9} : \frac{12}{8} - \frac{36}{7}

 

= 9 : \frac{3}{2} - \frac{36}{7} = \frac{18}{3} - \frac{36}{7} = 6 - \frac{36}{7} = \frac{42 - 36}{7} = \frac{6}{7}

 

Solution of exercise 7

Calculate:

1 16 ^ {\frac{3}{2}} = \sqrt{16^3} = \sqrt{(2^4)^3} = \sqrt {2^{12}} = 2^6 = 64

 

8 ^ {\frac{2}{3}} = \sqrt[3]{8^2} = \sqrt[3] {(2^3)^2} = \sqrt[3]{2^6} = 4

 

3   81^ {0.75} = 81^ {\frac{75}{100}} = 81^ {\frac{3}{4}} = \sqrt[4]{81^3} = \sqrt[4]{(3^4)^3} = \sqrt[4]{3^{12}} = 3^3 = 27

 

4   8 ^ {0.333...} = 8 ^ {\frac{3}{9}} = 8 ^ {\frac{1}{3}} = \sqrt[3]{2^3} = 2

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Emma

I am passionate about travelling and currently live and work in Paris. I like to spend my time reading, gardening, running, learning languages and exploring new places.