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LAWS OF EXPONENTS

LAWS OF EXPONENTS

The following laws of exponents are for multiplying and dividing monomials.

PRODUCT RULE:

a^m · aⁿ = a^m+n (when multiplying like bases, add the powers)

Examples:

1) x⁴ · x⁵ = x⁴⁺⁵= x⁹ 2) 5⁵ · 5⁸= 5⁵⁺⁸ = 5¹³

3) a⁷ · a · a¹² = a⁷⁺¹⁺¹²= a²⁰ 4) (3x⁶)(2x⁴) = (3·2)x⁶⁺⁴= 6x¹⁰

5) (4m⁸n²)(-2mn⁴)(5m⁴n³) = (4·-2·5)(m⁸⁺¹⁺⁴)(n²⁺⁴⁺³) = -40m¹³n⁹

POWER RULE:

(a^mb^m)ⁿ = a^mnb^mn
(when taking a monomial to a power, multiply the powers including the coefficient)

Examples:

1) (a⁴b³)² = a⁸b⁶ 2) (3m²n⁵)⁴ = 3⁴m⁸n²⁰ = 81m⁸n²⁰

3) (-2xy⁷z²)⁵= (-2)⁵x⁵y³⁵z¹⁰ = -32x⁵y³⁵z¹⁰

4) (6a⁹b⁶)²= 6²a¹⁸b¹²= 36a¹⁸b¹²
(-c⁴d²)⁵ (-1)⁵c²⁰d¹⁰-c²⁰d¹⁰

QUOTIENT RULE:

a^m = a^{m -
n} (when dividing with like bases, subtract the powers)
aⁿ
(Note: it is always the numerator's power minus the denominator's power)

Examples:

1) x⁶ = x^{6 – 4} = x² 2) m⁵n⁷= m^5-4n^7-10 = mn^-3

x ⁴ m⁴n¹⁰

3) a³b⁷ = a^{3 - (-5)}b^7-9 = a⁸n^-2

a^-5b⁹

ZERO POWER RULE:

a⁰ = 1 (any term to the zero power is one)

Examples:

1) (m⁵n⁷)⁰ = 1 2) (4m⁸n²)(-2mn4)⁰= (4m⁸n²)(1) = 4m⁸n²

****It is improper to leave negative powers in your final answer. All final answers should be written with positive powers. Therefore, you will need the following property. ****

NEGATIVE POWER RULE:

a^-n = 1 and 1 = aⁿ (take the reciprocal of the variable to the negative power)

aⁿa^-n

Examples:

1) 3x^-4 = 3 · 1 = 3 2) -5m^-8n²= -5y⁵n² x⁴x⁴ x¹⁰y^-5m⁸x¹⁰

NOTE: Apply the negative power rule to only negative POWERS.

EXAMPLES: Simplify the following expressions. Write the final answers without negative exponents.

Simplify means to combine like terms using the laws of exponents. Also, you may work with negative powers as you are simplifying within the problem. You just cannot leave negative powers in the final answer.

1) 12^-4(12⁸) = 12^-4+8 =12⁴ (product rule)

or = 1 •12⁸(negative power rule)

12⁴

= 12^{8 - 4} = 12⁴(quotient rule)

2) (5a⁴b⁶)(12abc)⁰(-2a²bc⁵) = (5a⁴b⁶) (1) (-2a²bc⁵) (zero power rule)
-10a⁶b⁷c⁵ (product rule)

3) -42m⁶n^-3p⁵ = -7m^6-11n^-3-(-5)p^5-5 (quotient rule)

6m¹¹n^-5p⁵

= -7m^-5n²p⁰

= -7n² (negative power and zero power rule)
m⁵

4) (-3ab⁶ )⁵= (-3)⁵a⁵b³⁰ (power rule)

(-a⁵b²)⁷ (-1)⁷a³⁵b¹⁴

= -243a⁵b³⁰
-1a³⁵b¹⁴

= 243a^5-35b^30-14 (quotient rule)

= 243a^-30b¹⁶

= 243b¹⁶ (negative power rule)
a³⁰

5) 2m⁷n³ • (3mp⁸)³ = 2m⁷n³ • 27m³p²⁴ (power rule)

(3n⁵p^-3)² 2mn⁶9n⁵p^-6 2mn⁶

= 54m¹⁰n³p²⁴ (product rule)

18mn¹¹p^-6

= 3 m^10-1n^3-11 p^{24 - (-6)} (quotient rule)

= 3m⁹n^-8p³⁰

= 3m⁹p³⁰ (negative power rule)

n⁸

6) (7xy)(-x⁴y³)⁵(2x⁵y⁶)^-2 = (7xy)(-1⁵x²⁰y¹⁵)(2^-2x ^-10y^-12) (power rule)

= (7xy)(-x²⁰y¹⁵)(x ^-10y^-12) (negative power rule)
2²

= -7 x^1+20-10y^1+15-12 (product rule)

= -7 x¹¹y⁴
4

7) (2)^-5 = (need to apply the negative power rule first before you can multiply)

= or

= is the final answer.

8) = ÷ (quotient rule)

= 5¹⁰ ÷ 5^-13

= 5^10-(-13) (quotient rule)

= 5²³

9) = (power rule)

= = (product rule)

= x^{12m +
12 – (8m – 6 )} = x^{12m + 12 – 8m + 6} (quotient rule)

= x^{4m + 18}

DMU Timestamp: July 29, 2014 21:38