By I. Kaplansky
An algebraic prelude Continuity of automorphisms and derivations $C^*$-algebra axiomatics and simple effects Derivations of $C^*$-algebras Homogeneous $C^*$-algebras CCR-algebras $W^*$ and $AW^*$-algebras Miscellany Mappings holding invertible parts Nonassociativity Bibliography
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5 3 1a b. 3 ,y 7 0 A 32y Solution 3 3 3 5 5 # 2a 2 5 2a2 5 2a2 a. = = = 3 3 3 2 3 3 a 1 a 1 a 2 a 2 a 3 # 3 2 3 3 • Use 1 a 2a = 2 a = a. (continued) 28 CHAPTER P PRELIMINARY CONCEPTS b. 3 A 32y = 16y 13 13 13 # 12y = = = 8y 132y 412y 412y 12y Try Exercise 112, page 30 To rationalize the denominator of a fractional expression such as 1 1m + 1n we use the conjugate of 1m + 1n, which is 1m - 1n. The product of these conjugate pairs does not involve a radical. ( 1m + 1n)( 1m - 1n) = m - n EXAMPLE 10 Rationalize the Denominator Rationalize the denominator.
Recall that a = a1. (x4y 3)(xy 5z 2) = x4 + 1y 3 + 5z 2 = x 5y8z 2 • Add the exponents of the like bases. a7b a5 = a7 - 2b 1 - 5 = a 5b-4 = 4 2 5 ab b • Subtract the exponents of the like bases. # # (uv 3) 5 = u 1 5v 3 5 = u 5v 15 a 5 3 1#3 5#3 • Multiply the exponents. 3 15 15 2x 2 x 2x 8x b = 1 # 3 4 # 3 = 3 12 = 4 5y 5 y 5y 125y 12 Question • Can the exponential expression • Multiply the exponents. x 5y 3 be simplified using the properties of exponents? Integrating Technology c Exponential expressions such as a b can be confusing.
V = 240 meters per second (speed of a commercial jet) c. 0 * 107 meters per second (10% of the speed of light) Sun d. 5 * 108 meters per second (50% of the speed of light) 135. 023 * 1023 atoms. Find the mass of one hydrogen atom. e. 7 * 108 meters per second (90% of the speed of light) 136. 0078t ). a. How much digoxin remains in the blood of a patient 4 hours after taking a 2-milligram dose? b. M. M. M.? f. Use your answers from a. through e. En is adequate for most of our common experiences involving motion (walking, running, bicycling, driving, flying, and so on).