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Calculate these roots: a) [3]1000 - [3]125, b) [3]64, c) sqrt(25 - 16), d) sqrt(100 - 64). Compare these calculations. Replace with >, < or =. a) 23 32

Mathematics

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Here are the solutions to your math problems:

5. Calculate these roots:
(Note: The instruction says "cube roots", but the symbols for c) and d) are square roots. I will solve a) and b) as cube roots and c) and d) as square roots, following the notation shown.)

  • a) 10003−1253\sqrt[3]{1000} - \sqrt[3]{125}
  1. Calculate the cube roots.

    $\sqrt[3]{1000} = 10$  
    $\sqrt[3]{125} = 5$
    
  2. Subtract the results.

    $10 - 5 = 5$  
    The answer is $\boxed{5}$.
    
    • b) 643\sqrt[3]{64}
  3. Find the number that, when multiplied by itself three times, equals 64.

    $4 \times 4 \times 4 = 64$  
    The answer is $\boxed{4}$.
    
    • c) 25−16\sqrt{25 - 16}
  4. Perform the subtraction inside the square root.

    $25 - 16 = 9$
    
  5. Calculate the square root of the result.

    $\sqrt{9} = 3$  
    The answer is $\boxed{3}$.
    
    • d) 100−64\sqrt{100 - 64}
  6. Perform the subtraction inside the square root.

    $100 - 64 = 36$
    
  7. Calculate the square root of the result.

    $\sqrt{36} = 6$  
    The answer is $\boxed{6}$.
    

    6. Compare these calculations. Replace □\square with >>, << or ==.

    • a) 23□322^3 \square 3^2
  8. Calculate 232^3.

    $2^3 = 2 \times 2 \times 2 = 8$
    
  9. Calculate 323^2.

    $3^2 = 3 \times 3 = 9$
    
  10. Compare the values.

    $8 < 9$  
    The comparison is $\boxed{2^3 < 3^2}$.
    
    • b) 2×4□242 \times 4 \square 2^4
  11. Calculate 2×42 \times 4.

    $2 \times 4 = 8$
    
  12. Calculate 242^4.

    $2^4 = 2 \times 2 \times 2 \times 2 = 16$
    
  13. Compare the values.

    $8 < 16$  
    The comparison is $\boxed{2 \times 4 < 2^4}$.
    
    • c) 8×8□828 \times 8 \square 8^2
  14. Calculate 8×88 \times 8.

    $8 \times 8 = 64$
    
  15. Calculate 828^2.

    $8^2 = 8 \times 8 = 64$
    
  16. Compare the values.

    $64 = 64$  
    The comparison is $\boxed{8 \times 8 = 8^2}$.
    
    • d) 92□9×29^2 \square 9 \times 2
  17. Calculate 929^2.

    $9^2 = 9 \times 9 = 81$
    
  18. Calculate 9×29 \times 2.

    $9 \times 2 = 18$
    
  19. Compare the values.

    $81 > 18$  
    The comparison is $\boxed{9^2 > 9 \times 2}$.
    
    • e) 72□737^2 \square 7^3
  20. Calculate 727^2.

    $7^2 = 7 \times 7 = 49$
    
  21. Calculate 737^3.

    $7^3 = 7 \times 7 \times 7 = 343$
    
  22. Compare the values.

    $49 < 343$  
    The comparison is $\boxed{7^2 < 7^3}$.
    
    • f) 83□273\sqrt[3]{8} \square \sqrt[3]{27}
  23. Calculate 83\sqrt[3]{8}.

    $\sqrt[3]{8} = 2$
    
  24. Calculate 273\sqrt[3]{27}.

    $\sqrt[3]{27} = 3$
    
  25. Compare the values.

    $2 < 3$  
    The comparison is $\boxed{\sqrt[3]{8} < \sqrt[3]{27}}$.
    
    • g) 63□546^3 \square 5^4
  26. Calculate 636^3.

    $6^3 = 6 \times 6 \times 6 = 216$
    
  27. Calculate 545^4.

    $5^4 = 5 \times 5 \times 5 \times 5 = 625$
    
  28. Compare the values.

    $216 < 625$  
    The comparison is $\boxed{6^3 < 5^4}$.
    

    7. Mr Tladi wants to pave his verandah.

    • a) If the length of his verandah is 4 m4 \text{ m} and the breadth is 3 m3 \text{ m}, how many m2\text{m}^2 of bricks does he need?
  29. Identify the formula for the area of a rectangle.

    $\text{Area} = \text{length} \times \text{breadth}$
    
  30. Substitute the given values.

    $\text{Area} = 4 \text{ m} \times 3 \text{ m}$
    
  31. Calculate the area.

    $\text{Area} = 12 \text{ m}^2$  
    Mr Tladi needs $\boxed{12 \text{ m}^2}$ of bricks.
    
    • b) Mr Tladi needs sand to do his paving. If the truck measures 3 m3 \text{ m} by 2 m2 \text{ m} by 1 m1 \text{ m}, how many m3\text{m}^3 of sand can it hold?
  32. Identify the formula for the volume of a rectangular prism.

    $\text{Volume} = \text{length} \times \text{breadth} \times \text{height}$
    
  33. Substitute the given dimensions of the truck.

    $\text{Volume} = 3  m \times 2
    
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