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UNIT 2 Powers and roots.

1. Powers

1.1. Exponents with negative bases

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Exercise1. Calculate

Exercise 2. Fill in the gaps

Exercise 3. Calculate

Exercise 4. Find the sign of this powers (only the sign) (-1)12 (-8)501 (-6)204

1.2. Powers of 1 and 0

1.3. Operations with powers of the same base

Product

Exercise 5. Fill in the gaps

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Division

Exercise 6. Fill in the gaps

Power of a power

Exercise 7. Fill in the missing numbers

Exercise 8. Express as just one power a) (−3)8 ∙ (−3)2 ∙ (−3)7 b) [(−29)]3 ∙ (−2) c) −(−3)5 ∙ (−3)2 d) (−9)4∙(−9)2

(−9)6

e) [(−57)]2 ∙ [(−53)]5 f) (−6)2 ∙ (−6)5 ∙ (−6)

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1.4. Operations with powers of the same exponent

Product When doing the product of two powers with the same exponent, the exponent remains unchanged and the bases are multiplied Example: 53*83= 403

Division When doing the division of two powers with the same exponent, the exponent remains unchanged and the bases are divided

Example: 353:53= 73 (−27)6

(3)6= (−27)6: (3)6 = (−9)6

Exercise 9. Express as just one power

Exercise 10. Express as just one power

Exercise 11. Find the value of a

Exercise 11. Fill in the gaps

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2. Roots

Exercise 12.Calculate the following square roots:

Exercise 13.Estimate the value of the following square roots:

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Exercise 14.Calculate the integer square root and the remainder

2.1. Calculating the square root of a number

Example: Find √645

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Exercise 15.Calculate the integer square root and the remainder

2.2. Operations with square roots

2.2.1. Multiplication

The square root of a product is equal to the product of the square roots of each factor

√𝐚 ∙ √𝐛 = √𝐚 ∙ 𝐛 √𝟐𝟐 ∙ √𝟏𝟏 = √𝟐𝟐 ∙ 𝟏𝟏

2.2.2. Division The square root of a division is equal to the division of the square roots of each factor √𝐚:√𝐛 = √𝐚:𝐛 √𝟑𝟏:√𝟗 = √𝟑𝟏:𝟗

2.2.3. Power of the square root

�√𝒂�

𝒎= √𝒂𝒎 �√𝟑𝟏�

𝟐= √𝟑𝟏𝟐

Exercise 16. Find the value of a

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Exercise 17. Express as a power

3. Order of operations (PEDMAS)

Exercise 18. Calculate

Exercise 19. Calculate

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Exercise 20. Calculate

Exercise 21. Write as a product of roots and calculate

Exercise 22. Write as a power

Exercise 23. Calculate

Exercise 24. Write each number as a power in two different ways

Exercise 25. Write as a single power

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Exercise 26. Find the value of a

Exercise 27. Calculate

Exercise 28. Write all the perfect square between 200 and 300. (A perfect square is a number that can be expressed as the product of two equal integers) Exercise 29. Find the integer square root and the remainder of:

a) 36 b) 13 c) 92 d) 58 e) 140 Exercise 30. The integer square root of a number is 15 and the remainder is 40. Find the number. Exercise 31. Calculate

Exercise 32. Express as a single root and calculate

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Exercise 33. Express as products and divisions of roots and calculate.

Exercise 34. Express as a product of two roots and calculate.

Exercise 35. Factorise the radicand and calculate √𝟗𝟗𝟗 Exercise 36. (Difficult) Calculate.

Exercise 37. (Difficult) Express the radicand as the square of a number and calculate.

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Exercise 38. Calculate

Exercise 39. (Difficult) Calculate

Exercise 40. What is the fewest number of years that must elapse from the year 2007 to be a perfect square. Exercise 41. Peter wants to fence a square plot of 1225 square meters. How many meters of wire are needed?