Laws Of Exponents

LAWS OF EXPONENTS

Product Rule -multiplying powers with same bases

• a

m

•

a =a n

m+n

5 • 5 = (5•5)(5•5•5) 2

3

=5

2+3

=5

5

Quotient Rule -dividing powers with same bases

2. a = a m

m–n

, a≠0

a 5 5-2 5 / 5 = 5•5•5•5•5 = 5 3 5•5 =5 n 2

Examples:

• x •x = 12 3 4 11 2. m n • m n = 6 9 3. 2 b = 2 2 2b 8 15 4 12 4. 7 c / 7 c = 5

7

Examples:

5. s t • s t = 10 3 s t 13 6

5 8

6. a b • a c d = 2 12 3 2 2 a b cd 5 15 3 8 4 12 7. q r • q r /q r = 11 7

14 4 9

Zero Exponent Rule

3. a = a m

m–m

= a , a≠0 0

a 4 4 4-4 7 / 7 = 7•7•7•7 = 7 7•7•7•7 0 = 7 =1 m

Negative Exponent Rule

4. a

–m

= 1 a

m

, a≠0

3 = 3 • 3 • 3 • 3 =3 1 7 3 3•3•3•3•3•3•3 = 3 3 4

4–7

Examples:

• p •p = 0 –4 4 –1 2. t u • t u = 9 0 3. 3 s = 12 –2 3 s -3 4 4 –6 4. 5 k / 5 k = 5

–7

Examples:

5. a b • a b = –2 2 a b –3 – 6

5 8

6. m n •2 m n = –5 0 –5 2 3 2 m n •m n –2 – 7 – 4 8 –6 0 7. p s • p s /p s = –5

6

–2

–3 –5

Power of Power

5. (a ) = a 2 3 (5 ) = (5•5)(5•5)(5•5) m n

mn

=5•5•5•5•5•5=5

= 5 •5 •5 5 2

2

2=

1+1+1+1+1+1

2+2+2

=5

6

Power of Product

6. (ab) = a b 3 3 (3b) = (3•b) = (3•b)(3•b)(3•b) 1+1+1 1+1+1 =3•3•3•b•b•b=3 •b 3 3= 3 = 3 •b 27b m

m m

Power of Quotient

7. a

m

=a

m

, b≠0

b b 3 2s = (2s)(2s)(2s) t t•t•t 3 3 3 = 2•2•2•s•s•s= 2 s = 8s 3 3 t•t•t t t m

Examples:

• (2a b ) = 2 4 4 2. (p / q ) = 2 5 3 3. 3 m = 3 2 3m 3 4 4 6 3 4. (4 r / 4 r ) = 2 3 3

Examples:

5. c d • 2 c 3 2 2 2cd 2 6

5 7 4

-

6. x y •2 x y 5 5 2 3 2 xy • x y 2 7 5 2 7 3 7. h k • h k / h k = 5

6

2

3 5

3

Examples:

• (q • r ) = 0 –3 3 3 4 2. (w x • w x ) = 2 –3 5 3. 3 s 4 –2 3s –2 4 –2 –6 2 4. (5 k / 5 k ) = 5

–7

3

Examples:

5. a b • a b –2 2 a b –3 – 6

5 8

–3

-

6. t v •2 t v –4 0 –3 5 7 2 t v •t v –4 –3 –2 0 –6 -3 4 7.(c d • c d /c d ) –6

0

– 4 – 2 –4 2