Exercise 1

Calculate:

2^7 : 2^6

5^7 : 5^3

2^5 \cdot 2^4 \cdot 2

3^3 \cdot 3^4 \cdot 3

(5 \cdot 2 \cdot 3)^4

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Exercise 2

Calculate:

(5^3)^4

(3^4)^4

(2^2)^4

(2^5)^4

Exercise 3

Calculate:

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

(8^2)^3 = [(2^3)^2]^3 = (2^6)^3

(9^3)^2 = [(3^2)^3]^2 = (3^6)^2

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

(27^2)^5 = [(3^3)^2]^5 = (3^6)^5

6 (4^3)^2 = [(2^2)^3]^2 = (2^6)^2

(4 \cdot 2 \cdot 3)^4

Exercise 4

Calculate:

(-2)^2 \cdot (-2)^3 \cdot (-2)^4 = (-2)^9

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

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

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

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

6 2^2 : 2^{-3} = 2^5

2^{-2} : 2^{-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:

16^{\frac{3}{2}}

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

81^{0.75}

8^{0.333...}

 

Solution of exercise 1

Calculate:

2^7 : 2^6 = 2

5^7 : 5^3 = 5^4 = 625

2^5 \cdot 2^4 \cdot 2 = 2^{10} = 1024

3^3 \cdot 3^4 \cdot 3 = 3^8 = 6561

(5 \cdot 2 \cdot 3)^4 = 30^4 = 8.1  10^5

 

Solution of exercise 2

Calculate:

(5^3)^4 = 5^{12}

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

(2^2)^4 = 2^8

(2^5)^4 = 2^{20}

 

Solution of exercise 3

[(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}

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

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

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

(4 \cdot 2 \cdot 3)^4 = 24^4

 

Solution of exercise 4

Calculate:

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

(-2)^-2 \cdot (-2)^3 \cdot (-2)^4 = (-2)^5 = -32

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 = \frac{1}{32}

6 2^2 : 2^{-3} = 2^5 = 32

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

 

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

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

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

 

Solution of exercise 6

Calculate:

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} = 2^2 = 4

81^{0.75} = 82^{\frac{75}{100}} = 82^ {\frac{3}{4}} = \sqrt[4]{81^3} = \sqrt[4] {(3^4)^3} = \sqrt[4]{3^{12}} = 3^3 = 27

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