Exponents Power 5 exponent 3 base Example: 125 53 means that 53 is the exponential form of the number 125. 53 means 3 factors of 5 or 5 x 5 x 5 The Laws of Exponents: #1: Exponential form: The exponent of a power indicates how many times the base multiplies itself. x x x x x x x x n n times n factors of x Example: 5 5 5 5 3 #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 2 2 2 6 3 63 2 9 512 #3: Dividing Powers: When dividing Powers with the same base, KEEP the BASE & SUBTRACT the EXPONENTS! m x m n mn x x x n x So, I get it! When you divide Powers, you subtract the exponents! 6 2 6 2 4 2 2 22 16 Try these: 12 1. 3 3 2 2 7. 2. 52 54 8. 3. a a 5 2 4. 2s 4s 2 7 12 8 9. 5. (3) (3) 2 3 6. s t s t 2 4 7 3 s 4 s 9 3 5 3 s t 4 4 st 5 8 10. 36a b 4 5 4a b SOLUTIONS 2 2 4 5 2 a 1. 3 3 3 3 81 2 4 6 2 4 2. 5 5 5 5 2 2 3. a a a 5 2 4. 2s 4s 2 4 s 2 7 5. (3) (3) (3) 2 3 6. s t s t 2 4 7 3 s 7 2 7 2 3 8s (3) 243 2 7 43 t 9 5 s t 9 7 SOLUTIONS 12 7. 8. 9. 10. s 12 4 8 s s 4 s 9 3 9 5 4 3 3 81 5 3 12 8 s t 12 4 8 4 8 4 s t s t 4 4 st 5 8 36a b 5 4 8 5 3 36 4 a b 9 ab 4 5 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 32 (5 ) 5 3 2 5 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 2 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 2 2 16 4 81 3 3 4 Try these: 5 2 5 1. 3 3. 2a 4. 2 a b 2. a s 7. t 2 39 8. 5 3 3 4 2 3 2 5 3 2 5. (3a ) 2 2 2 4 3 6. s t 2 st 9. 4 rt 5 8 2 36a b 10. 4 5 4a b 8 SOLUTIONS 2 5 1. 3 2. a 3 4 10 3 a12 2a 3. 2a 2 2 3 3 5 3 2 4. 2 a b 23 8a 6 22 52 32 4 10 6 10 6 2 a b 2 a b 16 a b 5. (3a ) 3 a 22 9a 4 2 2 2 2 4 3 6. s t 23 43 s t s t 6 12 SOLUTIONS 5 s 7. t 5 s 5 t 2 3 8. 5 34 3 9 2 3 8 2 4 2 2 8 st st s t 9. 4 2 r rt r 8 36a b 10 4 5 4 a b 5 8 2 9ab 3 2 92 a 2b32 81a 2b6 #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 1 m x 1 1 5 3 5 125 and 3 1 2 3 9 2 3 #8: Zero Law of Exponents: Any base powered by zero exponent equals one. x 1 0 So zero factors of a base equals 1. That makes sense! Every power has a coefficient of 1. 50 1 and a0 1 and (5 a ) 0 1 Try these: 1. 2a b 0 2 y 2 y 4 2. a 5 1 3. 2 4. s 4s 7 s t 2 5. 3x y 6. 3 4 2 4 0 1 2 7. x 2 39 8. 5 3 2 2 s t 9. 4 4 s t 2 5 36a 10. 4 5 4a b 2 2 SOLUTIONS 0 1. 2a b 1 2 1 3. a 5 a 5 2 7 4. s 4s 4s 5 1 2 5. 3x y 2 4 0 6. s t 3 4 3 x y 1 4 8 12 8 x 81y12 SOLUTIONS 1 2 7. x 9 2 3 8. 5 3 2 1 x 4 4 x 3 2 4 2 1 3 8 3 8 s t 2 2 2 4 4 s t 9. 4 4 s t s t 10 2 5 b 2 2 10 36a 9 a b 2 10. 4 5 81a 4 a b 2 2