Exponent Laws MPM2D: Principles of Mathematics Consider the - PDF document

p o l y n o m i a l s p o l y n o m i a l s Exponent Laws MPM2D: Principles of Mathematics Consider the expression x 2 · x 3 . Using the definition of exponentiation, x 2 · x 3 can be expressed as ( x · x )( x · x · x ) = x · x · x · x · x = x 5 . More generally, x a · x b = ( x · x · . . . · x ) · ( x · x · . . . · x ) � �� � � �� � Exponent Laws a times b times = x a + b . = x · x · . . . · x � �� � a+b times J. Garvin Product of Like Powers Law For any real, non-zero values a , b and x , x a · x b = x a + b . If the bases are not the same, this rule does not apply. The expression 2 4 · 3 2 cannot be simplified further. J. Garvin — Exponent Laws Slide 1/15 Slide 2/15 p o l y n o m i a l s p o l y n o m i a l s Exponent Laws Exponent Laws Next, consider the expression x 3 � x 3 � 2 . Now, consider x 2 . x 3 � 2 becomes ( x · x · x ) · ( x · x · x ) = x 6 . � Rewriting, Rewriting, x 3 x 2 can be expressed as x · x · x = x . In general, ( x a ) b = x · x = x ab . ( x · x · . . . · x ) ( x · x · . . . · x ) · ( x · x · . . . · x ) · . . . · ( x · x · . . . · x ) � �� � � �� � � �� � � �� � More generally, x a a times a times a times a times = x a − b . x b = = x · x · . . . · x � �� � ( x · x · . . . · x ) � �� � b times a-b times � �� � b times Power of a Power Law For any real, non-zero values a , b and x , ( x a ) b = x ab . Quotient of Like Powers Law For any real, non-zero values a , b and x , x a x b = x a − b . Like the earlier Product Law, the bases must be the same. J. Garvin — Exponent Laws J. Garvin — Exponent Laws Slide 3/15 Slide 4/15 p o l y n o m i a l s p o l y n o m i a l s Exponent Laws Exponent Laws Consider ( xy ) 2 next. Like the power of a product, the power of a quotient can be similarly defined. In its longer form, ( xy ) 2 = ( xy )( xy ) = ( x · x )( y · y ) = x 2 y 2 . � x � 2 � x � � x � = x 2 In general, ( xy ) a = ( xy ) · ( xy ) · . . . · ( xy ) = For instance, = · y 2 . y y y � �� � a times � x � a � x � � x � � x � = x a y a . = x a ( x · x · . . . · x ) · ( y · y · . . . · y ) In general, = · · . . . · � �� � � �� � y a y y y y a times a times � �� � a times Power of a Product Law For any real, non-zero values a , x and y , ( xy ) a = x a y a . Power of a Quotient Law � x � a = x a For any real, non-zero values a , x and y , y a . y J. Garvin — Exponent Laws J. Garvin — Exponent Laws Slide 5/15 Slide 6/15

p o l y n o m i a l s p o l y n o m i a l s Exponent Laws Exponent Laws What about the expression x 0 ? Example According to the Quotient Law, x a Simply the expressions x 4 · x 7 , z 8 � x � 5 x a = x a − a = x 0 . � k 3 � 5 , (2 p ) 3 and z 6 , . 2 At the same time, k k = 1, as long as k � = 0. x 4 · x 7 = x 4+7 = x 11 . z 8 k = x a If k = x a , then k z 6 = z 8 − 6 = z 2 . x a = x 0 = 1. k 3 � 5 = k 3 × 5 = k 15 . � Zero Exponent Law (2 p ) 3 = 2 3 p 3 = 8 p 3 . For any real, non-zero value of x , x 0 = 1. = x 5 2 5 = x 5 � x � 5 32. 2 J. Garvin — Exponent Laws J. Garvin — Exponent Laws Slide 7/15 Slide 8/15 p o l y n o m i a l s p o l y n o m i a l s Exponent Laws Exponent Laws What does a negative exponent, like x − 2 , mean? Example Since x x 3 = x 1 − 3 = x − 2 , and since x x 3 = 1 x 2 , then x − 2 = 1 Evaluate 1 234 567 0 . x 2 . Since the base is non-zero, 1 234 567 0 = 1. In general, x a · x − a = x a +( − a ) = x 0 = 1, assuming x � = 0. Therefore, x a · x − a = 1, which can be rearranged to Example x − a = 1 Express x − 4 using positive exponents. x a . x − 4 = 1 Negative Exponent Law x 4 . Again, x cannot equal zero. For any real, non-zero value of x and any real, positive value of a , x − a = 1 x a . J. Garvin — Exponent Laws J. Garvin — Exponent Laws Slide 9/15 Slide 10/15 p o l y n o m i a l s p o l y n o m i a l s Exponent Laws Exponent Laws Sometimes it is necessary to combine two or more exponent Example laws to simplify an expression. Simplify (5 p − 3 q ) − 2 , using positive exponents. Example Simplify x 5 y 3 x 2 y 7 , using positive exponents. (5 p − 3 q ) − 2 = 5 − 2 p ( − 3)( − 2) q − 2 = 1 5 2 · p 6 · 1 q 2 x 5 y 3 p 6 x 2 y 7 = x 5 − 2 y 3 − 7 = 25 q 2 = x 3 y − 4 = x 3 y 4 J. Garvin — Exponent Laws J. Garvin — Exponent Laws Slide 11/15 Slide 12/15

p o l y n o m i a l s p o l y n o m i a l s Scientific Notation Scientific Notation Scientific notation is a system, used in many sciences, that Example expresses numbers using powers of 10. Express 75 328 143 using scientific notation, to two decimal For example, the number 352 can be expressed as 3 . 52 × 10 2 , places. since 3 . 52 × 10 2 = 3 . 52 × 100 = 352. Shifting the decimal 7 places to the left, and rounding down, It is often used as a shorthand notation for very small or very 75 328 143 = 7 . 53 × 10 7 . large numbers. Example For instance, 3 800 000 000 000 (3 trillion, 800 billion) can be Express 0 . 000 031 874 using scientific notation, to two expressed more simply as 3 . 8 × 10 12 . decimal places. By convention, scientific notation expresses all numbers with one digit before the decimal point – that is, 4 . 3 × 10 3 rather Shifting the decimal 5 places to the right, and rounding up, than 43 × 10 2 . 0 . 000 031 874 = 3 . 19 × 10 − 5 . Positive exponents indicate the decimal point has been shifted left, while negative exponents indicate a right shift. J. Garvin — Exponent Laws J. Garvin — Exponent Laws Slide 13/15 Slide 14/15 p o l y n o m i a l s Questions? J. Garvin — Exponent Laws Slide 15/15