Laws of Exponents

1. EXPONENTS AND ITS LAWS

2. EXPONENTIAL FORM
• Is an alternative method of expressing number.
For example:
𝑥2
, 22
, 23
, 52

3. 52 EXPONENT
BASE
Is a number which
represents the number of
times a base is to be
multiplied by itself.
Is a number which is to
be multiplied by itself
according to the
exponent.
𝟓𝟐
= 5  5 = 25

4. Law 1: Product Law
(𝑎𝑚) 𝑎𝑛 = 𝑎𝑚+𝑛
States that when multiplying two powers with the same base, just add
the exponents.

5. Example:
1. (23) 22 = 23+2 = 25 = 32
2. (𝑥5
) 𝑥4
= 𝑥5+4
= 𝑥9
3. (32) 34 = 32+4 = 36 = 729

6. You try!
1. (24) 25 =
2. (𝑥32
) 𝑥25
=
3. (𝑦59
) 𝑦51
=

7. Law 2: Quotient Law
𝑎𝑚
𝑎𝑛
= 𝑎𝑚−𝑛
States that when dividing two powers with the same base, just subtract
the exponents.

8. Example:
1.
27
23 = 27−3 = 24 = 16
2.
35
33 = 35−3
= 32
= 9
3.
28
26 = 28−6 = 22 = 4

9. You try!
1.
43
42 =
2.
𝑥20
𝑥13 =
3.
𝑦105
𝑦87 =

10. Law 3: Power of a power
𝑥𝑚 𝑛 = 𝑥𝑚𝑛
To simplify any power of power, simply multiply the exponents.

11. Example:
1. 23 2 = 2(3)(2) = 26 = 64
2. 𝑥4 3 = 𝑥(4)(3) = 𝑥12
3. 32 2 = 3(2)(2) = 34 = 81

12. You try!
1. 𝑎2 5 =
2. 22 3
=
3. 𝑥100 3 =

13. Law 4: Power with different bases
𝑏𝑐 𝑛 = 𝑏𝑛𝑐𝑛
𝑎
𝑏
𝑛
=
𝑎𝑛
𝑏𝑛
In dividing different bases, it can’t be simplified unless the exponents
are equal.

14. Example:
1. 2𝑥 2 = 22𝑥2 = 4𝑥2
2. 5𝑥𝑦 3
= 53
𝑥3
𝑦3
= 125𝑥3
𝑦3
3.
𝑥
3
2
=
𝑥2
32 =
𝑥2
9
4.
4
𝑎
2
=
42
𝑎2 =
16
𝑎2

15. You try!
1. 4𝑎𝑏 2 =
2. 10𝑥𝑦𝑧 2
=
3.
𝑥
𝑦
4
=
4.
2
4
2
=

16. Law 5: Zero Exponent
𝑎0 = 1
A nonzero base raised to zero the result is equal to one.
For example:
30 = 1
𝑥0 = 1
3𝑥
𝑦
0
= 1

17. You try!
1. 7, 645, 321 0
=
2. 30 + 𝑥0 + 3𝑦 0 =

18. Law 6: Negative Exponent
𝑎−𝑛 =
1
𝑎𝑛
A nonzero base raised to a negative exponent is equal to the reciprocal
of the base raised to the positive exponent.

19. Example:
1. 3−2 =
1
32 =
1
9
2. 2−8 =
1
28 =
1
256
3.
2
5
−6
=
5
2
6
=
56
26 =
15,625
64

20. You try!
1. 𝑥−2 =
2. 3−3
=
3. 5 − 3 −2
=

21. Law 1: Product Law (𝑎𝑚) 𝑎𝑛 = 𝑎𝑚+𝑛
Law 2: Quotient Law
𝑎𝑚
𝑎𝑛 = 𝑎𝑚−𝑛
Law 3: Power of a power 𝑥𝑚 𝑛 = 𝑥𝑚𝑛
Law 4: Power with different bases 𝑏𝑐 𝑛 = 𝑏𝑛𝑐𝑛 𝑎
𝑏
𝑛
=
𝑎𝑛
𝑏𝑛
Law 5: Zero Exponent

22. Let’s simplify!
1. (𝑥10) 𝑥12 =
2. (𝑦−3) 𝑦8 =
3. 𝑚15 3 =
4. 𝑑−3 2
=
5.
𝑏8
𝑏12 =
6. 𝑎−4 −4 =
7.
𝑧23
𝑧15 =
8.
𝑐3
𝑐−2 =
9.
𝑥7𝑦10
𝑥3𝑦5 =
10.
𝑎8𝑏2𝑐0
𝑎5𝑏5 =