Lesson 10: Proofs of Laws of Exponents
We have learned several rules of exponents:
- xᵐ × xⁿ = xᵐ⁺ⁿ
- (xᵐ)ⁿ = xᵐⁿ
- xᵐ ÷ xⁿ = xᵐ⁻ⁿ
- (x/y)ⁿ = xⁿ / yⁿ
- x⁻ⁿ = 1/xⁿ
Now, we will prove that the rules always work, even when exponents are zero or negative.
Exercises and Solutions
1. Show that (x/y)ⁿ = xⁿ / yⁿ works for n = 0.
Example: (5/7)⁰ = ?
Left side: (5/7)⁰ = 1
Right side: 5⁰ / 7⁰ = 1/1 = 1
Both sides are equal
2. Show that (x⁻¹)ⁿ = x⁻ⁿ.
Example: (3⁻¹)² = ?
Left side: (3⁻¹)² = (1/3)² = 1/9
Right side: 3⁻² = 1/3² = 1/9
Both sides are equal
3. Prove that the power rule (xᵐ)ⁿ = xᵐⁿ works even when m is negative.
Let m = −2, n = 3
Left side: (x⁻²)³ = (1/x²)³ = 1/x⁶
Right side: x⁻²×³ = x⁻⁶ = 1/x⁶
Both sides are equal
4. Prove that xᵐ × xⁿ = xᵐ⁺ⁿ is always true.
Example: x³ × x⁻⁵ = ?
Left side: x³ × x⁻⁵ = x³ × (1/x⁵) = x³/x⁵ = 1/x²
Right side: x³⁺⁻⁵ = x⁻² = 1/x²
Both sides are equal
5. A communication chain problem
You send a message to 7 people. Each sends it to 5 people, and each of them sends to 5 more, and again to 5 more. How many people get the message (not counting you)?
First group: 7
Second group: 7×5 = 35
Third group: 35×5 = 175
Fourth group: 175×5 = 875
Total = 7 + 35 + 175 + 875 = 1092
6. Show that (1.27³⁶)⁸⁵ = 1.27³⁰⁶⁰.
Multiply exponents: 36×85 = 3060
So result = 1.27³⁰⁶⁰
7. Show that (2/13)¹²⁷ × (2/13)⁵⁶ = (2/13)¹⁸³.
Same base, add exponents: 127 + 56 = 183
Result = (2/13)¹⁸³
8. Show that x¹²⁷ × x⁵⁶ = x¹⁸³.
Same base, add exponents: 127 + 56 = 183
9. Write 105×92 ÷ 64 as prime factors with exponents.
105 = 3×5×7
92 = 2²×23
64 = 2⁶
Expression = (3×5×7×2²×23) ÷ 2⁶ = 3×5×7×23×2⁻⁴
10. Prove that (2³)⁴ = 2¹² by writing out the factors.
(2³)⁴ = (2×2×2)(2×2×2)(2×2×2)(2×2×2) = 12 twos multiplied = 2¹²
11. Expand (x²)³.
(x²)³ = (x²)(x²)(x²) = x²⁺²⁺² = x⁶
12. Prove that (3²×4³)² = 3⁴×4⁶.
(3²×4³)² = (3²)²×(4³)² = 3⁴×4⁶
13. Show that (x/y)² = x² / y².
(x/y)² = (x/y)(x/y) = (x×x)/(y×y) = x²/y²
14. Simplify (5²)⁻¹.
(5²)⁻¹ = 1/5² = 1/25
15. Simplify (x⁻³)².
= x⁻⁶ = 1/x⁶
16. Simplify (a⁻²b³)².
= a⁻⁴b⁶ = b⁶/a⁴
17. Simplify (2³×3²)⁻¹.
= 1/(2³×3²) = 1/72
18. Expand (x²y³z)³.
= x²×³ y³×³ z³ = x⁶y⁹z³
19. Show that (x⁰)ⁿ = 1.
x⁰ = 1 so (x⁰)ⁿ = 1
20. Show that (1/x)ⁿ = x⁻ⁿ.
(1/x)ⁿ = 1/xⁿ = x⁻ⁿ