Exponent Laws With Numerical Bases MPM1D: Principles of Mathematics - PDF document

a l g e b r a a l g e b r a Exponent Laws With Numerical Bases MPM1D: Principles of Mathematics Recap � 3 � 5 3 × 5 9 Simplify, then evaluate, . 5 4 Use the product, quotient and power of a power rules. Exponent Laws (Variable Bases) � 3 � 3 � 5 3 × 5 9 � 5 12 = J. Garvin 5 4 5 4 � 5 8 � 3 = = 5 24 J. Garvin — Exponent Laws (Variable Bases) Slide 1/17 Slide 2/17 a l g e b r a a l g e b r a Exponent Laws With Variable Bases Exponent Laws With Variable Bases In all of the previous examples we have dealt with powers Consider two simple powers with a common variable base, such as x 2 and x 3 . with a numerical base, like 2 3 . According to the rules of exponentiation, x 2 = x · x and An algebraic power may use a variable base instead, and may x 3 = x · x · x . include a leading numerical value known as a coefficient . For example, the power 3 x 5 has a variable x , an exponent 5, If we were to find their product, we would obtain the and a coefficient 3. following. The power x 4 y 3 has two variables, x and y , with exponents 4 x 2 x 3 and 3 respectively. x 2 · x 3 = ���� � �� � x · x · x · x · x Since no coefficient is specified, it is assumed that it has a = x 5 value of 1. Note that x 5 = x 2+3 , showing that the product rule holds for variable bases. J. Garvin — Exponent Laws (Variable Bases) J. Garvin — Exponent Laws (Variable Bases) Slide 3/17 Slide 4/17 a l g e b r a a l g e b r a Exponent Laws With Variable Bases Exponent Laws With Variable Bases Example All three of the previously-explored exponent laws can be used with powers having variable bases. Simplify x 8 · x 11 . Exponent Laws Using the product rule, x 8 · x 11 = x 8+11 = x 19 . The product rule, quotient rule and power of a power rule can be applied to powers with variable bases. Example • product rule: x m · x n = x m + n � x 5 � 6 . Simplify • quotient rule: x m x n = x m − n x 5 � 6 = x 5 · 6 = x 30 . � Using the power of a power rule, • power of a power rule: ( x m ) n = x m · n J. Garvin — Exponent Laws (Variable Bases) J. Garvin — Exponent Laws (Variable Bases) Slide 5/17 Slide 6/17

a l g e b r a a l g e b r a Exponent Laws With Variable Bases Exponent Laws With Variable Bases Consider the expression 2 x 3 · 4 x 2 . Example � 8 � x 7 · x 9 Since multiplication is commutative , we can rearrange the Simplify values. x 12 2 x 3 · 4 x 2 = 2 · 4 · x 3 · x 2 Use all three exponent laws. = 8 x 5 � 8 � 8 � x 7 · x 9 � x 16 = The same is true for division. x 12 x 12 � x 4 � 8 6 x 7 2 · x 7 2 x 4 = 6 = x 4 = x 32 = 3 x 3 J. Garvin — Exponent Laws (Variable Bases) J. Garvin — Exponent Laws (Variable Bases) Slide 7/17 Slide 8/17 a l g e b r a a l g e b r a Exponent Laws With Variable Bases Exponent Laws With Variable Bases Example This suggests that the product and quotient rules are valid for powers involving coefficients, provided we multiply or Simplify 5 x 2 · 7 x 10 . divide the coefficients as necessary. Using the product rule, Product/Quotient Rules w/ Coefficients/Multiple 5 x 2 · 7 x 10 = 5 · 7 · x 2 · x 10 Variables = 35 x 12 When using the product/quotient rules with like-base powers involving coefficients, the new coefficient has a value equal to Example the product/quotient of the given coefficients. Simplify 18 x 9 10 x 5 . Using the quotient rule, 18 x 9 10 · x 9 10 x 5 = 18 x 5 = 9 5 x 4 J. Garvin — Exponent Laws (Variable Bases) J. Garvin — Exponent Laws (Variable Bases) Slide 9/17 Slide 10/17 a l g e b r a a l g e b r a Exponent Laws With Variable Bases Exponent Laws With Variable Bases Consider the expression (2 x ) 3 . Example � 3 x 7 � 4 . Using the definition of exponentiation, we can rewrite the Simplify expression in its longer form. (2 x ) 3 = (2 x )(2 x )(2 x ) Apply the exponent to the coefficient and to the variable. = (2 · 2 · 2)( x · x · x ) 3 x 7 � 4 = 3 4 x 7 · 4 � = 2 3 x 3 = 81 x 28 = 8 x 3 Like the other exponent laws, this can be extended to This suggests the following rule. examples involving more than one variable. Power of a Power Rule w/ Coefficients/Multiple Variables When using the power of a power rule with a power involving a coefficient, the new coefficient has a value equal to the given coefficient raised to the given power. J. Garvin — Exponent Laws (Variable Bases) J. Garvin — Exponent Laws (Variable Bases) Slide 11/17 Slide 12/17

a l g e b r a a l g e b r a Exponent Laws With Variable Bases Exponent Laws With Variable Bases Example Example � 6 x 5 y � 8 x 3 y 5 � 2 . � 2 Simplify Simplify . 9 x 2 y 3 Apply the exponent to the coefficient and to both variables. One method is to apply the exponent to all coefficients and 8 x 3 y 5 � 2 = 8 2 x 3 · 2 y 5 · 2 � variables, then simplify after. = 64 x 6 y 10 � 6 x 5 y � 2 = 6 2 x 10 y 2 9 x 2 y 3 9 2 x 4 y 6 = 36 x 10 y 2 81 x 4 y 6 = 4 x 6 9 y 4 J. Garvin — Exponent Laws (Variable Bases) J. Garvin — Exponent Laws (Variable Bases) Slide 13/17 Slide 14/17 a l g e b r a a l g e b r a Exponent Laws With Variable Bases Exponent Laws With Variable Bases A better method is to simplify first, since the values will be Example smaller and (possibly) easier to work with. � 2 x 3 � 4 Simplify (3 x 6 ) 2 . � 6 x 5 y � 2 � 2 x 3 � 2 = 9 x 2 y 3 3 y 2 Use the power of a power rule, and the quotient rule. = 2 2 x 6 3 2 y 4 � 2 x 3 � 4 (3 x 6 ) 2 = 2 4 x 3 · 4 = 4 x 6 3 2 x 6 · 2 9 y 4 = 16 x 12 Both methods lead to the same answer, but the latter is 9 x 12 preferred. = 16 9 Note that the variable x cancels completely. J. Garvin — Exponent Laws (Variable Bases) J. Garvin — Exponent Laws (Variable Bases) Slide 15/17 Slide 16/17 a l g e b r a Questions? J. Garvin — Exponent Laws (Variable Bases) Slide 17/17