This document provides rules and examples for exponents. The key rules are: 1) The product rule - when multiplying the same bases, add the exponents. 2) The quotient rule - when dividing the same bases, subtract the exponents. 3) Any number to the zero power is 1. 4) When raising a power to another power, multiply the exponents. The document also covers expanding powers, negative exponents, and cautions about order of operations with exponents. Practice problems with solutions are provided.

1. EXPONENT RULES & PRACTICE
1. PRODUCT RULE: To multiply when two bases are the same, write the base and ADD the exponents.
࢞࢓
· ࢞࢔
ൌ ࢞࢓ା࢔
Examples:
A. ‫ݔ‬ଷ
· ‫ݔ‬଼
ൌ ‫ݔ‬ଵଵ
B. 2ସ
· 2ଶ
ൌ 2଺
C. ሺ‫ݔ‬ଶ
‫ݕ‬ሻሺ‫ݔ‬ଷ
‫ݕ‬ସሻ ൌ ‫ݔ‬ହ
‫ݕ‬ହ
2. QUOTIENT RULE: To divide when two bases are the same, write the base and SUBTRACT the exponents.
࢞࢓
࢞࢔ ൌ ࢞࢓ି࢔
Examples:
A.
௫ఱ
௫మ ൌ ‫ݔ‬ଷ
B.
ଷఱ
ଷయ ൌ 3ଶ
C.
௫మ௬ఱ
௫௬య ൌ ‫ݕݔ‬ଶ
3. ZERO EXPONENT RULE: Any base (except 0) raised to the zero power is equal to one.
࢞૙
ൌ ૚
Examples:
A. ‫ݕ‬଴
ൌ 1 B. 6଴
ൌ 1 C. ሺ7ܽଷ
ܾିଵሻ଴
ൌ 1
4. POWER RULE: To raise a power to another power, write the base and MULTIPLY the exponents.
ሺ࢞࢓ሻ࢔
ൌ ࢞࢓·࢔
Examples:
A. ሺ‫ݔ‬ଷሻଶ
ൌ ‫ݔ‬଺
B. ሺ3ଶሻସ
ൌ 3଼
C. ሺ‫ݖ‬ହሻଶ
ൌ ‫ݖ‬ଵ଴
5. EXPANDED POWER RULE:
ሺ࢞࢟ሻ࢓
ൌ ࢞࢓
࢟࢔
ቀ
࢞
࢟
ቁ
࢓
ൌ
࢞࢓
࢟࢓
Examples:
A. ሺ2ܽሻଷ
ൌ 2ଷ
ܽଷ
ൌ 8ܽଷ
B. ሺ6‫ݔ‬ଷሻଶ
ൌ 6ଶሺ‫ݔ‬ଷሻଶ
ൌ 36‫ݔ‬଺
C. ቀ
௫మ
௬
ቁ
ସ
ൌ
൫௫మ൯
ర
௬ర ൌ
௫ఴ
௬ర
D. ቀ
ଶ௫
ଷ௬మቁ
ଷ
ൌ
ሺଶ௫ሻయ
ሺଷ௬మሻయ ൌ
ଶయ௫య
ଷయሺ௬మሻయ ൌ
଼௫య
ଶ଻௬ల
6. NEGATIVE EXPONENTS: If a factor in the numerator or denominator is moved across the fraction bar, the sign
of the exponent is changed.
࢞ି࢓
ൌ
૚
࢞࢓
૚
࢞ష࢓ ൌ ࢞࢓
ቀ
࢞
࢟
ቁ
ି࢔
ൌ ቀ
࢟
࢞
ቁ
࢔
Examples:
A. ‫ݔ‬ିଷ
ൌ
ଵ
௫య B. 4ିଶ
ൌ
ଵ
ସమ ൌ
ଵ
ଵ଺ C. െ4‫ݔ‬ହ
‫ݕ‬ିଶ
ൌ
ିସ௫ఱ
௬మ
D. ቀ
௫మ
௬
ቁ
ିଷ
ൌ ቀ
௬
௫మቁ
ଷ
ൌ
௬య
௫ల
E. ሺ3‫ݔ‬ିଶ
‫ݕ‬ሻሺെ2‫ݕݔ‬ିଷሻ ൌ െ6‫ݔ‬ିଵ
‫ݕ‬ିଶ
ൌ
ି଺
௫௬మ
F.
௔షమ௕య
௖షరௗషభ ൌ
௕య௖రௗ
௔మ G. ሺെ2‫ݔ‬ଶ
‫ݕ‬ିସሻିଶ
ൌ ቀ
ିଶ௫మ
௬ర ቁ
ିଶ
ൌ ቀ
௬ర
ିଶ௫మቁ
ଶ
ൌ
௬ఴ
ସ௫ర
CAUTION: െ‫ݔ‬ ്
ଵ
௫
For example: െ3 ്
ଵ
ଷ
REMEMBER: An exponent applies to only the factor it is directly next to unless parentheses enclose other factors.
Examples:
A. ሺെ3ሻଶ
ൌ ሺെ3ሻሺെ3ሻ ൌ 9 B. െ3ଶ
ൌ െ9

2. EXPONENTS PRACTICE
Simplify:
1. 3 · 4ଷ
2. 4‫ݔ‬ଷ
· 2‫ݔ‬ଷ
3. ‫ݔ‬ହ
· ‫ݔ‬ଷ
4. 2‫ݔ‬ଷ
· 2‫ݔ‬ଶ
5.
଺ఱ
଺య
6.
௫ర
௫ళ
7. 8଴
8. െሺ9‫ݔ‬ሻ଴
9. ሺ‫ݕ‬ସሻଷ
10. ሺ‫ݔ‬ଶ
‫ݕ‬ሻସ
11.
଺௫ళ
ଶ௫ర
12.
଼௫ఱ
ସ௫మ
13. ሺ2ܿ݀ସሻଶሺܿ݀ሻହ
14. ሺ2݂݃ସሻସሺ݂݃ሻ଺
15.
௫ఱ௬ల
௫௬మ
16.
௫మ௬ఱ
௫௬ర
17. ቀ
ସ௫ఱ௬
ଵ଺௫௬ర
ቁ
ଷ
18. ቀ
ହ௫య௬
ଶ଴௫௬ఱ
ቁ
ସ
19. ‫ݕ‬ି଻
20. 7ିଶ
21.
ଵ
௫షఱ
22.
ଵ
ଶషర
23. ‫ݔ‬ହ
· ‫ݔ‬ିଵ
24. ‫ݔ‬ି଺
25. ‫ݔ‬ଽ
· ‫ݔ‬ି଻
26. ሺ݆ିଵଷሻሺ݆ସሻሺ݆଺ሻ
27.
௫షభ
௫షఴ
28.
ହଶ௫ల
ଵଷ௫షళ
29. ݂ିଷሺ݂ଶሻሺ݂ିଷሻ
30.
௫షర
௫షవ
31.
ଶସ௫ల
ଵଶ௫షఴ
32.
ଷ௫మ௬షయ
ଵଶ௫ల௬య
33. ሺ2‫ݔ‬ଷ
‫ݕ‬ିଷሻିଶ
34.
ଶ௫ర௬షర
଼௫ళ௬య
35. ሺ4‫ݔ‬ସ
‫ݕ‬ିସሻଷ
36. 5‫ݔ‬ଶ
‫ݕ‬ሺ2‫ݔ‬ସ
‫ݕ‬ିଷሻ
37. ቀ
ି଻௔మ௕య௖బ
ଷ௔య௕ర௖య
ቁ
ିସ
38. ቀ
ିଶ௔య௕మ௖బ
ଷ௔మ௕య௖ళ
ቁ
ିଶ
EXPONENTS PRACTICE ANSWERS
1. 192
2. 8‫ݔ‬଺
3. ‫ݔ‬଼
4. 4‫ݔ‬ହ
5. 36
6.
ଵ
௫య
7. 1
8. െ1
9. ‫ݕ‬ଵଶ
10. ‫ݔ‬଼
‫ݕ‬ସ
11. 3‫ݔ‬ଷ
12. 2‫ݔ‬ଷ
13. 4ܿ଻
݀ଵଷ
14. 16݂ଵ଴
݃ଶଶ
15. ‫ݔ‬ସ
‫ݕ‬ସ
16. ‫ݕݔ‬
17.
௫భమ
଺ସ௬వ
18.
௫ఴ
ଶହ଺௬భల
19.
ଵ
௬ళ
20.
ଵ
ସଽ
21. ‫ݔ‬ହ
22. 16
23. ‫ݔ‬ସ
24.
ଵ
௫ల
25. ‫ݔ‬ଶ
26.
ଵ
௝య
27. ‫ݔ‬଻
28. 4‫ݔ‬ଵଷ
29.
ଵ
௙ర
30. ‫ݔ‬ହ
31. 2‫ݔ‬ଵସ
32.
ଵ
ସ௫ర௬ల
33.
௬ల
ସ௫ల
34.
ଵ
ସ௫య௬ళ
35.
଺ସ௫భమ
௬భమ
36.
ଵ଴௫ల
௬మ
37.
଼ଵ௔ర௕ర௖భమ
ଶସ଴ଵ
38.
ଽ௕మ௖భర
ସ௔మ