Lesson 7: Numbers in Exponential Form Raised to a Power

Today, we learn how to handle exponents raised to another power.

Rule:
(xᵐ)ⁿ = xᵐⁿ

That means when a power is raised to another power, we multiply the exponents.

Example:
(2³)⁴ = 2³×⁴ = 2¹²

Why?
Because (2³)⁴ means (2³)(2³)(2³)(2³) = 2³+³+³+³ = 2¹²

Exercises

1. (15³)⁹ = ?
Solution: Multiply exponents: 3 × 9 = 27
Answer: 15²⁷

2. (−2)⁵⁸ = ?
Solution: Multiply exponents: 5 × 8 = 40
Answer: (−2)⁴⁰

3. (3.417)⁴ = ?
Solution: No simplification, just write: (3.417)⁴

4. (17ᵇ)⁴ = ?
Solution: Multiply exponents: b × 4 = 4b
Answer: 17⁴ᵇ

5. Sarah wrote (3⁵)⁷ = 3¹². Correct her mistake.
Solution: (3⁵)⁷ = 3⁵×⁷ = 3³⁵, not 3¹².

6. A number satisfies 2⁴ = 256. What equation does x = 4 satisfy?
Solution: If x = 4, then 2ˣ = 16.
But 2⁴ = 16, not 256. To get 256, we need 2⁸.
So x must satisfy 2²ˣ = 256.

7. (11 × 4)⁹ = ?
Solution: Distribute exponent: 11⁹ × 4⁹

8. (3² × 7⁴)⁵ = ?
Solution: Multiply exponents: 3²×⁵ × 7⁴×⁵ = 3¹⁰ × 7²⁰

9. (3²a⁴)⁵ = ?
Solution: Multiply exponents: 3²×⁵ × a⁴×⁵ = 3¹⁰a²⁰

10. (5x)⁷ = ?
Solution: 5⁷x⁷

11. (5x²)⁷ = ?
Solution: 5⁷(x²)⁷ = 5⁷x¹⁴

12. (a²b³)⁴ = ?
Solution: a²×⁴ × b³×⁴ = a⁸b¹²

13. How is (x/y)ⁿ related to xⁿ and yⁿ?
Solution: (x/y)ⁿ = xⁿ / yⁿ

14. (2³)⁴ = ?
Solution: Multiply exponents: 2³×⁴ = 2¹²

15. (10²)⁵ = ?
Solution: Multiply exponents: 10¹⁰

16. (−7²)³ = ?
Solution: (−7²)³ = (−49)³ = −117,649

17. (−7)²³ = ?
Solution: Multiply exponents: 2 × 3 = 6 → (−7)⁶. Even exponent → positive.

18. (½)⁴ = ?
Solution: (½)⁴ = 1/16

19. (½)³² = ?
Solution: Multiply exponents: (½)³²

20. (2x³y²)⁴ = ?
Solution: Multiply exponents: 2⁴x³×⁴y²×⁴ = 16x¹²y⁸

21. (a²b)³ = ?
Solution: a⁶b³

22. (m³n⁵)² = ?
Solution: m³×²n⁵ײ = m⁶n¹⁰

23. (4³)² = ?
Solution: 4³×² = 4⁶ = 4096

24. (x²y³z⁴)² = ?
Solution: x⁴y⁶z⁸

25. (p²q³)⁴ = ?
Solution: p⁸q¹²

26. (3a²b³)² = ?
Solution: 3²a⁴b⁶ = 9a⁴b⁶

27. (2x²y)³ = ?
Solution: 2³x⁶y³ = 8x⁶y³