Exponents {5 exponent Power base Example: 125 = 53 means that 53 is the exponential form of the number 125 53 means factors of or x x The Laws of Exponents: #1: Exponential form: The exponent of a power indicates how many times the base multiplies itself x =1 x ×4 x ×4 x4××2 ××x4×x4×x43 ×x n n −times n factors of x Example: = ×5 ×5 #2: Multiplying Powers: If you are multiplying Powers with the same base, KEEP the BASE & ADD the EXPONENTS! x ×x = x m So, I get it! When you multiply Powers, you add the exponents! n m +n 26 × 23 = 6+3 = 29 = 512 #3: Dividing Powers: When dividing Powers with the same base, KEEP the BASE & SUBTRACT the EXPONENTS! m x m n m −n = x ÷ x = x n x So, I get it! When you divide Powers, you subtract the exponents! 6−2 = = 2 = 16 Try these: 12 × = 2 52 × 54 = a ×a = s × s = 12 (−3) × (−3) = s t ×s t = s = s = s t = 4 st 10 36a b = 4a b SOLUTIONS 2+ a ×a = a 5+ =a × = = = 81 2+ =5 × = 5 s × s = × × s (−3) × (−3) = (−3) s t ×s t = s 2+ 2+3 = 8s = (−3) = −243 2+ 4+3 t =s t SOLUTIONS 12 10 s 12 − s = s = s 9 −5 = = 81 = 12 s t 12 − 8− s t = s t = 4 st 36a b − −5 36 ÷ × a b = ab = 4a b #4: Power of a Power: If you are raising a Power to an exponent, you multiply the exponents! (x ) n m So, when I take a Power to a power, I multiply the exponents =x mn (5 ) = 3×2 =5 #5: Product Law of Exponents: If the product of the bases is powered by the same exponent, then the result is a multiplication of individual factors of the product, each powered by the given exponent ( xy ) So, when I take a Power of a Product, I apply the exponent to all factors of the product n = x ×y n n ( ab) = a b 2 #6: Quotient Law of Exponents: If the quotient of the bases is powered by the same exponent, then the result is both numerator and denominator , each powered by the given exponent n x  x  ÷= n y y  So, when I take a Power of a Quotient, I apply the exponent to all parts of the quotient n 4 16 2   = = 81 3 Try these: ( ) ( a 5 ) = = ( ) = ( a b ) a s   = t2  39    = 3  5 (−3a ) = 2 ( s t ) = =  st    =  rt   36a b   = 10    4a b  SOLUTIONS ( ) ( ) a = ( a12 = ( ) 2a 10 3 = a ) a b 2×3 = 8a = 2×2 a 5×2b 3×2 = a10b = 16a10b (−3a ) = ( − 3) × a 2×2 ( s t 12 2 ) 2×3 4×3 =s t =s t = 9a SOLUTIONS s   = t s t 3    = 34 3  ( ) =3 2  st    st s t   =   =  r  rt   r   36a b 10   4a b   = 9ab  ( ) 2 3×2 =9 a b = 81a b #7: Negative Law of Exponents: If the base is powered by the negative exponent, then the base becomes reciprocal with the positive exponent So, when I have a Negative Exponent, I switch the base to its reciprocal with a Positive Exponent Ha Ha! If the base with the negative exponent is in the denominator, it moves to the numerator to lose its negative sign! x −m = m x 1 = = 125 and −3 = =9 −2 #8: Zero Law of Exponents: Any base powered by zero exponent equals one x =1 So zero factors of a base equals That makes sense! Every power has a coefficient of 50 =1 and a =1 and (5a ) =1 Try these: ( 2a b ) = y × y −4 = (a ) = −2 −1 ( x y −2 (s t ) ) −4 = = −2 s t   4  = s t  −2  36a  10   =  4a b  2 s × s = −1 2    =  x −2  39    = 3  SOLUTIONS ( ) a b = 1 −2 −4 y × y = y = y −1 a = a −2 s × s = 4s ( ) ( x y −2 ( s t ) ) −4 = (3 x y = −4 −12 x ) = 81y12 SOLUTIONS −1 2     x  −2 3    3  −1 x 4 = x  =   ( ) = −2 ( −2 =3 = −8 ) s t  − − −2 4 =s t  4  = s t  s t  −2 10 b − − 10  36a  a b =   10   = 81 a  4a b  2