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Einstein Circle

This lesson includes a summary of squares, square roots, and the exponent laws. Additional topic lessons and related exercises are available in the math toolbox.

Welcome to the EXPONENTS instruction unit.


Exponents - Part A

Exponents - Part B

These instructional quizzes are short, limited to only about a dozen questions, and they are not graded or timed. Occasionally there will be a question introducing a topic that you are not expected to answer at first. Do not panic. The instructional material will likely be introduced in the following answer explanation. The purpose of these instructional modules is to review subject material by topic that you will likely encounter on the exam and provide some topic specific tips to improve your ability to solve related problems. 



Khan Academy (Exponents)

Exponents Introduction

Approximately 5 minutes video.

Exponents Rules

Approximately 15 minutes video.

Fractional and Negative Exponents

Approximately 15-30 minutes in several videos.

Part 1 begins on the Introduction lesson page. Each succeeding part is on the consecutive Lesson Lecture page.

The Story of Eratosthenes – Critical Thinking Adventure Part 4

The experiment would theoretically work based in a city or location not on one of the tropics at a longitude of 23.5º. The shadow in that case would be non-zero at noon on the solstice. The necessary measure is the difference of shadow angles measured between locations. It is the difference that corresponds to the angle from the center of earth to the two locations. The benefit of Syene at 24º longitude is that it did not require a measurement and would not add additional errors from the process of measuring, however, that assumes the sun is exactly overhead and it is now known there is a slight difference between 23.5° and 24º so there was some error introduced by using Syene in either case. At least the use of Syene allowed Eratosthene to never leave the luxury of Alexandria!

Would you prefer a location to the north or south or a location east/west or anything in between? In theory any direction could work as long as there was curvature of the earth and lightspeed communication available. Unfortunately, the the earth rotates and the sun therefore moves across an east-west trajectory in the sky and that would make an east-west design impractical. If the earth rotates one half circle or 180º in 12 hours that is 15º per hour or 1° every 4 minutes. That is a large portion of the angle being measured (~7º). Simultaneous measurements would have to be taken and how could that possibly be achieved with accuracy 500 miles apart in the absence of iPhones (lightspeed communication)? A north-south position would elimiinate the impact of the earth’s rotation and make measurements easier and more reliable.

Could you design locations closer or longer than 500 miles apart? What would be ideal? If the locations were too close then the instrument error measuring the angle would be too high relative to the arc length itself. e.g. perhaps the measurement error of the angle equated to ±10 miles distance on the earth’s surface. That would be 20% of a 50 mile distance between locations, but only 2% of a 500 mile distance. Longer would be preferable, but how exactly would you measure long distances in 245 B.C. with accuracy. GPS is out of the question!

There is a very good reason Eratosthene chose the 2nd location of Alexandria and it had little to do with the luxury of staying home. The nearly 500 mile distance was the longest well surveyed stretch of flat land (Egyptian surveying was among the best of the time and the distance between the two cities was measured annually) along a nearly meridional (north-south) line. Similar measurements were made a century later, but without an accurate distance between locations none stood up to the reliability of Eratosthenes’ results.

The entire experiment depends on a vertical column perpendicular to a surface that in theory represented an exact sphere. What if the measurements were against a miniature version of the leaning tower of pisa (not yet to exist for almost 2,000 years)? In other words, any small deviation from vertical would change the shadow length and the entire angle measurement. Even 1º is an unacceptable error. Could you design an experiment on the east coast near NYC or Washington with similar tools Eratosthenes had at his disposal? Why or why not? What equipment would you use if you were Eratosthenes to avoid this error?

Part 5 (the final part) begins on the Lesson 4: Lecture page.

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