  Assignment 45

 "Concerning Egypt, I shall extend my remarks to a great length, because there is no country that possesses so many wonders." -- (Herodotus, 5th century B.C.) Egypt represents a rich field for ancient historical and mathematical research. Among other things, the pyramids of Egypt required considerable mathematical and engineering skills, including knowledge of volume, area, estimation, right angles, and perhaps the relationship we know as the Pythagorean theorem to compute the size, shape, number, and arrangement of the stones used to build the pyramids. The Great Pyramid of Gizeh was built around 2600 B.C., some two thousand years before the historian Herodotus made the statement quoted above. In addition to Gizeh, 35 major pyramids stand near the Nile River in Egypt. Each was built to preserve the mummified body of an Egyptian king, which was placed in a secret chamber filled with gold and precious objects. Some scholars believe that pyramids, rather than structures of a different shape, were used for tombs because the pyramids' sloping sides paralleled the slanting rays of the sun, enabling the soul of the kings to climb to the sky and join the gods. Why did Herkimer attempt to dry his wet shoes by placing them on New York newspapers? Answer: He felt that these were the Times that dry men's soles. Things Herky would like to know: If you have one for the road at a party, are you likely to get a trooper for a chaser? Can a tattoo artist be accused of having designs on a client? ASSIGNMENT #45 Reading: Section 7.1, pages 401-404. Written: Nothing from the book. Study the mistakes you made on the semester exam. Learn from these mistakes. Mathematical word analysis:PERIMETER: From the Greek roots peri (around) and metron (measure). A perimeter is literally a measure around the borders of a figure.
This section introduces you to fractional exponents. Make sure you read it carefully. Once again, language becomes important. You must know what the symbolism means, and be able to translate the symbolism into everyday words.

Fractional exponents do "jive" with previously developed mathematical ideas. As a simple example, ũ4 can be written as 41/2. We know that (ũ4)(ũ4) = 4. What about (41/2)(41/2)? Well, in previous use of positive integer exponents, we learned, for instance, that x2x7 = x9. That is, we added the exponents in this example of products when the two numbers had the same base. If we do the same with the fractional exponents, we would obtain (41/2)(41/2) = 41/2+1/2 = 41 = 4. Hence, we do get "consistency" if we write ũ4 = 41/2 .

This definition is important: If m and n are positive integers, then

xm/n = nũxm

Some simple examples:

45/2 = ũ45 = ũ1024 = 32. Note that on your calculator, if you calculate 4^(5/2), you get 32. We can also note that 45/2 = (22)5/2 = 25 = 32. CAREFUL: If you calculate 4^5/2, you don't get 32... and you should know why!

272/3 = 3ũ 272 = 3ũ 729 = 9. Note that on your calculator, if you calculate 27^(2/3), you get 9. We can also note that 272/3 = (33)2/3 = 32 = 9.

Fractional exponents offer some tremendous advantages when one works in higher branches of mathematics, such as calculus.

 Problem: Evaluate 87/3. Solution (with communication): 87/3 = (23)7/3 = 27 = 128. Problem: Evaluate 32-2/5. Solution (with communication): 32-2/5 = (25)-2/5 = 2-2 = 1/22 = 1/4.