Exponent Rules & Practice

EXPONENT RULES & PRACTICE 1. PRODUCT RULE: To multiply when two bases are the same, write the base and ADD the exponents. Examples: x ^ m * x ^ n = x ^ (m + n) A. x ^ 3 * x ^ 8 = x ^ 11 B. 2 ^ 4 * 2 ^ 2 = 2 ^ 6 C. (x ^ 2 * y)(x ^ 3 * y ^ 4) = x ^ 5 * y ^ 5 2. QUOTIENT RULE: To divide when two bases are the same, write the base and SUBTRACT the exponents. (x ^ m)/(x ^ n) = x ^ (m - n) Examples: (x ^ 5)/(x ^ 2) = x ^ 3 8. (3 ^ 3)/(3 ^ 3) = 3 ^ 2 C. (x ^ 2 * y ^ 5)/(x * y ^ 3) = x * y ^ 2 3. ZERO EXPONENT RULE: Any base (except 0) raised to the zero power is equal to one. Examples: x ^ 0 = 1 A. y ^ 0 = 1 B. 6 ^ 0 = 1 C. (7a ^ 3 * b ^ - 1) ^ 0 = 1 4. POWER RULE: To raise a power to another power, write the base and MULTIPLY the exponents. (x ^ m) ^ n = x ^ (mn) Examples: A. (x ^ 3) ^ 2 = x ^ 6 B. (3 ^ 2) ^ 4 = 3 ^ 8 C. (z ^ 5) ^ 2 = z ^ 10 5. EXPANDED POWER RULE: (xy) ^ m = x ^ m * y ^ n (x/y) ^ m = (x ^ m)/(y ^ m) Examples: A. (2a) ^ 3 = 2 ^ 3 * a ^ 3 = 8a ^ 3 B. (6x ^ 3) ^ 2 = 6 ^ 2 * (x ^ 3) ^ 2 = 36x ^ 6 ((x ^ 2)/y) ^ 4 = ((x ^ 2) ^ 4)/(y ^ 4) = (x ^ y)/(y ^ 4) ((2x)/(3y ^ 2)) ^ 3 = ((2x) ^ 3)/((3y ^ 3) ^ 3) = (2 ^ 3 * x ^ 3)/(3 ^ 3 * (y ^ 2) ^ 3) = (8x ^ 3)/(27y ^ 6) 6. NEGATIVE EXPONENTS: If a factor in the numerator or denominator is moved across the fraction bar, the sign of the exponent is changed. x ^ (- m) = 1/(x ^ m) 1/(x ^ (- m)) = x ^ m (x/y) ^ (- n) = (y/x) ^ n Examples: x ^ - 3 = 1/(x ^ 3) B. 4 ^ - 2 = 1/(4 ^ 2) = 1/16 C. - 4x ^ 5 * y ^ - 2 = (- 4x ^ 5)/(y ^ 2) ((x ^ 2)/y) ^ - 3 = (y/(x ^ 2)) ^ 3 = (y ^ 3)/(x ^ 6) E. (3x ^ - 2 * y)(- 2x * y ^ - 3) = - 6x ^ - 1 * y ^ - 2 = - 6/(x * y ^ 2) F (a ^ - 2 * b ^ 3)/(c ^ - 4 * a ^ - 1) = (b ^ 3 * c ^ 4 * d)/(a ^ 2) G. (- 2x ^ 2 * y ^ - 4) ^ - 2 = ((- 2x ^ 2)/(y ^ 4)) ^ - 2 = ((y ^ 4)/(- 2x ^ 2)) ^ 2 = (y ^ 0)/(4x ^ 4) CAUTION: -x ne 1/x For example: -3 ne 1/3 REMEMBER: An exponent applies to only factor it is directly next to unless parentheses enclose other factors. Examples: A. (- 3) ^ 2 = (- 3)(- 3) = 9 B. - 3 ^ 2 = - 9 EXPONENTS PRACTICE Simplify: 1. 3 * 4 ^ 3 15. (x ^ 5 * y ^ 6)/(x * y ^ 2) 2. 4x ^ 3 * 2x ^ 3 16. (x ^ 2 * y ^ 5)/(x * y ^ 4) 3. x ^ 5 * x ^ 3 4. 2x ^ 3 * 2x ^ 2 17. ((4x ^ 5 * y)/(16x * y ^ 4)) ^ 3 28. (52x ^ 6)/(13x ^ - 7) 29. f ^ - 3 * (f ^ 2)(f ^ - 3) 30 (x ^ - 4)/(x ^ - 9) 5. (6 ^ 5)/(6 ^ 2) 18. ((5x ^ 3 * y)/(20x * y ^ 5)) ^ 4 31. (24x ^ 6)/(12x ^ - 6) 6 (x ^ 4)/(x ^ 7) 19. y ^ - 7 7. 8 deg 7 ^ - 2 8. - (9x) ^ 0 21 1/(x ^ (- pi)) 9. (y ^ 4) ^ 3 10. (x ^ 2 * y) ^ 4 22 1/(2 ^ - 4) . (8x ^ 5)/(4x ^ 2) 13. (2c * d ^ 4) ^ 2 * (cd) ^ 5 14. (2f * g ^ 4) ^ 4 * (fg) ^ 6 23. x ^ 5 * x ^ - 1 25. x ^ 9 * x ^ - 7 26. (j ^ - 13)(j ^ 4)(j ^ 6) 32. (3x ^ 2 * y ^ - 3)/(12x ^ 6 * y ^ 3) 33. (2x ^ 3 * y ^ - 3) ^ - 2 35. (4x ^ 4 * y ^ - 4) ^ 3 36. 5x ^ 2 * y(2x ^ 4 * y ^ - 3) 37. ((- 7a ^ 2 * b ^ 3 * c ^ 0)/(3a ^ 3 * b ^ 4 * c ^ 3)) ^ - 4 38. ((- 2a ^ 3 * b ^ 2 * c ^ 0)/(3a ^ 2 * b ^ 3 * c ^ 7)) ^ - 2 EXPONENTS PRACTICE ANSWERS 1. 192 16. xy 2. 8x ^ 6 17. (x ^ 12)/(64y ^ 9) 30. 3. 18 (x ^ 10)/(256y ^ 16) 31. 2x ^ 14 4. 4x ^ 5 32. 1/(4x ^ 4 * y ^ 6) 5. 36 19 1/(y ^ 7) 6. 1/(x ^ 3) 20 1/49 7. 1 21. 34. 1/(4x ^ 3 * y ^ 7) 8.-1 22. 16 9. y ^ 12 23. 10. x ^ 8 * y ^ 4 36. (10x ^ 6)/(y ^ 2) 25. x ^ 2 37. (81a ^ 4 * b ^ 4 * c ^ 12)/2401 12. 2x ^ 3 26. 1/(j ^ 3) 13. 4c ^ 7 * d ^ 13 38. (9b ^ 2 * c ^ 14)/(4a ^ 2) 14. 16f ^ 10 * g ^ 22 154.4 28. 4x ^ 13 (2x ^ 4 * y ^ - 4)/(8x ^ 3 * y ^ 3) (6x ^ 7)/(2x ^ 4) 1/(f ^ 4) (y ^ 6)/(4x ^ 6) (64x ^ 12)/(3 ^ 42) 1/(x ^ 6) x ^ - 6 3x ^ 3 (x ^ - 1)/(x ^ (- n))