# Section 10-1: 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 power-of-ten eponent is positive. If the count is to the left, the power-of-ten 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 power-of-ten 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 powers-of-ten. 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 power-of- factor. Position caret (new decimal position) so that power will be a multiple of three. Write with power-of- factor. Write in engineering notation using metric prefies. 8 Ω Change to engineering notation with power-of- factor.. 8 Ω Change power-of- factor to appropriate prefi. 8. ΜΩ

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