Section 101: Laws of Exponents


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1 Section : Laws of Eponents Learning Outcome Multiply:  ( ) =   = = To multiply like bases, add eponents, and use common base. Rewrite answer with positive eponent. Learning Outcome Write the reciprocals of,, and 7. The reciprocal of is ; the reciprocal of is ; the reciprocal of 7 is 7. Write the following with positive eponents and as ordinary numbers. = = = =, Divide: =  = = To divide like bases, subtract eponents, and use common base. Rewrite answer with positive eponent. Learning Outcome ( y ) = y  8 ( a ) = a = 8 a ( )( ) = ( )( ) = 0 ( ) = 0 = To simplify a power to a power, multiply eponents. Multiply eponents, then rewrite answer with a positive eponent. Simplify indicated power to power, then multiply like bases by adding eponents. Multiply eponents. Any base with a zero eponent is equal to.
2 ( ) ( ) 8 ( y ) ( ) ( y ) y 8 Raise both numerator and deominator to the power. Perform operations. Raise each factor to the power. Perform operations. Section : Polynomials Learning Outcome Identify each of the following algebraic epressions according to the number of terms in it: ; 7 + ; ; + monomial; A monomial is a polynomial that has only one term. 7 + trinomial; A trinomial is a polynomial that has three terms. + binomial; A binomial is a polynomial that has two terms. polynomial; A polynomial has only terms with positive integral eponents. Learning Outcome What is the degree of each polynomial? ; 7 + ; ; + Degree of term is ; degree of monomial is. 7 + Degree of each term is,, and 0, respectively. Degree of the trinomial is. Degree of each term is and 0, respectively. Degree of the binomial is. + Degree of each term is,,, and 0, respectively. Degree of the polynomial is.
3 Learning Outcome Arrange the polynomial in descending powers of : Eamine the degree of each term. List term with highest eponent first and so on. Arrange the polynomial in ascending powers of : Eamine the degree of each term. List term with highest eponent first and so on. Note that a constant has a degree of zero. Section : Basic Operations with Polynomials Learning Outcome Regroup terms that have the same base and eponent. ( ) + ( ) Combine like terms. Terms with unlike eponents cannot be combined. Learning Outcome Simplify: ( y )( y z) ( y )( y z) = y z Multiply coefficients; multiply like bases by adding eponents. Learning Outcome 8 y Simplify: y 8 y y 8 y y = y = y Divide coefficients; divide like bases by subtracting eponents. Write final answer with positive eponents. Section : Powers of Ten and Scientific Notation Learning Outcome Multiply: ( )( 8 ) ( )( 8 ) = Use rule for multiplying like bases: Add eponents and keep the same base. Divide: 7 = = = Divide by subtracting eponents and keeping the same 7 base. Rewrite the result with a positive eponent.
4 Learning Outcome Write.8 as an ordinary number. To multiply by a power of with a positive integral eponent, shift the decimal to the right as many places as indicated by the eponent:.8 = 8,000 Write 8. as an ordinary number. To multiply by a power of with a negative integral eponent, shift the decimal to the left as many places as indicated by the eponent: 8. = Learning Outcome Write 0 in scientific notation. 0. =. Place a caret where the decimal will be to form a number greater than or equal to one, but less than. Then count from the caret to the original decimal. If the count is to the right, the poweroften eponent is positive. If the count is to the left, the poweroften eponent is negative. Write 0.0 in scientific notation. 00. = 00. =. Distance from caret to decimal is two places to left, so eponent of is . Learning Outcome Multiply and write result in scientific notation: (. 78 )(.8 ) (. 78 ) (.8 ) Multiply numerical factors and poweroften factors. = (..8)( 78 ) = (.8)( 8 ) Place caret so number factor is greater than or equal to one, but less than ten. 8 = (. 8)( ) Rewrite number factor in scientific notation. = (.8)( )( 8 ) Multiply powers of ten. = (.8)( 8 ) Divide and write result in scientific notation. (8. ) (.8 ) (8. ) (.8 ) (8. ) (.8 ) Rewrite division of ordinary numbers and division of powersoften. 8. = 8. = 0.8 Perform division. = 08. Place caret in ordinary number so factor is greater than or equal to one but less than ten. = 8. ( )( ) Write number in scientific notation. Multiply powers of ten. = 8.
5 Learning Outcome Write in engineering notation., Position caret (new decimal position) so that power will be a multiple of three. Write with powerof factor. Position caret (new decimal position) so that power will be a multiple of three. Write with powerof factor. Write in engineering notation using metric prefies. 8 Ω Change to engineering notation with powerof factor.. 8 Ω Change powerof factor to appropriate prefi. 8. ΜΩ
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