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1. Powers of Real Numbers(Exponents)

2. Exponents represent repeated multiplication. For example, Introduction

3. More generally, for any non-zero real number a and for any whole number n, Introduction In the exponential expression an, a is called the base and n is called the exponent.

4. a2 is read as ‘a squared’. a3 is read as ‘a cubed’. a4 is read as ‘a to the fourth power’. ... an is read as ‘a to the nth power’.

5. Caution!

6. Caution!

7. Properties of Exponents

8. Example:

9. Properties of Exponents

10. Example:

11. Properties of Exponents Homework.

12. Example:

13. Some Definitions of Exponents

14. Properties of Exponents Homework.

15. Example:

16. Properties of Exponents Homework.

17. Example:

18. Properties of Exponents Homework.

19. Example:

20. Properties of Exponents All powers of a positive real number a are positive, i.e. for a ∈ R, a > 0, and n ∈ Z, an > 0. 2. The even powers of a negative real number a are positive, i.e. for a ∈ R, a ≠ 0 and n ∈ Z, (–a)n = an (if n is an even number). 3. The odd powers of a negative real number a are negative, i.e. for a ∈ R, a ≠ 0 and n ∈ Z, (–a)n= –an (if n is an odd number)

21. Example:

22. Example:

23. Properties of Exponents The terms of an expression which have the same base and the same exponent are called like terms. We can add or subtract like terms. (x ⋅ an) + (y ⋅ an) + (z ⋅ an) = (x + y + z) ⋅ an (a ≠ 0)

24. Example:

25. Let a ∈ R – {–1, 0, 1} (a is a real number other than –1, 0 and 1). If am = an then m = n. Exponential Equations

26. 2x = 16 3x+1 = 81 22x + 1 = 8x – 1 Examples:

27. Exercises: