Lesson 6: Multiplication of Numbers in Exponential Form

When we multiply numbers that have the same base, we can add their exponents.

Rule:
xᵐ × xⁿ = xᵐ⁺ⁿ

Example:
2³ × 2⁴ = (2 × 2 × 2) × (2 × 2 × 2 × 2)
= 2 × 2 × 2 × 2 × 2 × 2 × 2
= 2⁷

So instead of multiplying everything, we just add the exponents (3 + 4 = 7).

Exercises

1. 14²³ × 14⁸ = ?
Solution: Same base (14). Add exponents: 23 + 8 = 31.
Answer: 14³¹

2. (−72)¹⁰ × (−72)¹³ = ?
Solution: Same base (−72). Add exponents: 10 + 13 = 23.
Answer: (−72)²³

3. 59⁴ × 59⁸ = ?
Solution: Same base (59). Add exponents: 4 + 8 = 12.
Answer: 59¹²

4. (−3)⁹ × (−3)⁵ = ?
Solution: Same base (−3). Add exponents: 9 + 5 = 14.
Answer: (−3)¹⁴

5. a²³ × a⁸ = ?
Solution: Same base (a). Add exponents: 23 + 8 = 31.
Answer: a³¹

6. b¹⁰ × b¹³ = ?
Solution: Same base (b). Add exponents: 10 + 13 = 23.
Answer: b²³

7. 9⁴ × 9⁷ = ?
Solution: Same base (9). Add exponents: 4 + 7 = 11.
Answer: 9¹¹

8. (−3)⁹ × (−3)ˣ = (−3)¹⁴. What is x?
Solution: 9 + x = 14 → x = 5.
Answer: x = 5

9. 9⁴ × 9⁶ × 9¹³ = ?
Solution: Same base (9). Add exponents: 4 + 6 + 13 = 23.
Answer: 9²³

10. 2³ × 2⁵ × 2⁷ × 2⁹ = ?
Solution: Same base (2). Add exponents: 3 + 5 + 7 + 9 = 24.
Answer: 2²⁴

11. 6⁵ × 4⁹ × 4³ × 6¹⁴ = ?
Solution: Different bases (6 and 4). We only combine like bases.
= 6⁵ × 6¹⁴ × 4⁹ × 4³
= 6¹⁹ × 4¹²

12. (−4)² × 17⁵ × (−4)³ × 17⁷ = ?
Solution: Combine like bases.
(−4)² × (−4)³ = (−4)⁵
17⁵ × 17⁷ = 17¹²
Answer: (−4)⁵ × 17¹²

13. 15² × 7² × 15 × 7⁴ = ?
Solution: Combine like bases.
15² × 15¹ = 15³
7² × 7⁴ = 7⁶
Answer: 15³ × 7⁶

14. 2⁴ × 8² = ?
Solution: Write 8 as 2³.
= 2⁴ × (2³)² = 2⁴ × 2⁶ = 2¹⁰

15. 3⁷ × 9 = ?
Solution: Write 9 as 3².
= 3⁷ × 3² = 3⁹

16. 5⁴ × 2¹¹ = ?
Solution: Different bases, cannot combine.
Answer: 5⁴ × 2¹¹

17. (2x³)(17x⁷) = ?
Solution: Multiply numbers and combine x’s.
= 2×17 × x³ × x⁷ = 34x¹⁰

18. a(a+b) = ?
Solution: Use distributive property.
= a² + ab

19. (a+b)² = ?
Solution: Expand binomial.
= a² + 2ab + b²

20. 7⁹ ÷ 7⁶ = ?
Solution: Same base. Subtract exponents: 9−6=3.
Answer: 7³

21. (−5)¹⁶ ÷ (−5)⁷ = (−5)⁹

22. (8/5)⁹ ÷ (8/5)² = (8/5)⁷

23. 13⁵ ÷ 13⁴ = 13¹ = 13

24. x⁵ ÷ x⁴ = x

25. (−2)⁷ × 95⁵ ÷ (−2)⁵ × 95⁴ = (−2)² × 95

26. (3x⁸)(−4x⁶) = −12x¹⁴

27. (5x)(11x⁴) = 55x⁵

28. 2,000,000,000 × 2,000,000,000,000 = 2⁹ × 10²¹ = 2¹ × 10²¹ = 4 × 10²¹