Lesson 5: Exponential Notation

When we multiply the same number many times, instead of writing it out again and again, we use exponential notation.

Example:
5 × 5 × 5 × 5 = 5⁴
Here, 5 is the base, and 4 is the exponent.
It means “multiply 5 by itself 4 times.”

General Rule:
xⁿ = x × x × … × x (n times)

Examples

1. 4 × 4 × 4 × 4 × 4 × 4 × 4 = ?
Solution: 7 fours multiplied → 4⁷

2. 3.6 × 3.6 × 3.6 × 3.6 × 3.6 × 3.6 × 3.6 = ?
Solution: 7 threes point six multiplied → 3.6⁷

3. (−11.63) × (−11.63) × … × (−11.63) (34 times) = ?
Solution: Base is −11.63, repeated 34 times → (−11.63)³⁴

4. 12 × 12 × … × 12 = ?

^{15 times}
Solution: 12¹⁵

5. (−5) × (−5) × … × (−5) = ?

^{10 times}
Solution: (−5)¹⁰

6. (7/2) × (7/2) × … × (7/2)= ?

^{21 times}
Solution: (7/2)²¹

7. (−13) × (−13) × … × (−13) = ?

^{6 times}
Solution: (−13)⁶

8. (−1/14) × (−1/14) × … × (−1/14) = ?

^{10 times}
Solution: (−1/14)¹⁰

9. x × x × … × x = ?

^{185 times}
Solution: x¹⁸⁵

10. x × x × … × x = ?

^{n times}
Solution: xⁿ

11a. (−1) × (−1) × … × (−1)

^{12 times}

Exponent is even → positive. Answer: +1

11b. (−1) × (−1) × … × (−1)

^{13 times}
Exponent is odd → negative. Answer: −1

12a. (−5)⁹⁵ → 95, odd exponent → negative

12b. (−1.8)¹²² → 122, even exponent → positive

13. If n is even, (−55)ⁿ is positive.
If n is odd, (−72.4)ⁿ is negative.

14. Josie said (−15)⁶ = −15⁶.
She is wrong because the exponent is even → answer is positive.
Correct: (−15)⁶ > 0

15. 9 × 9 × 9 × 9 = 9⁴

16. (−2) × (−2) × (−2) = (−2)³

17. (3/5) × (3/5) × (3/5) × (3/5) × (3/5) = (3/5)⁵

18. (−7)²⁰ → even exponent → positive

19. (−7)²¹ → odd exponent → negative

20. 2¹⁰ means multiply 2 by itself 10 times:
2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2