[Monte314]

I am teaching an 8-week Numeracy class to kids (ages 14 - 17) this Summer ("Numeracy" is to numbers as "Literacy" is to written language). The purpose of the class is to teach students a collection of practical skills that will enable to them to perform numerical calculations quickly, confidently, and accurately; to develop skills that facilitate mental computation; and to master useful number facts.

One of our topics for today was using simple number facts to be able to quickly and accurately estimate expressions involving square, cube, and fourth roots of decimal values. The ability to rapidly estimate such expressions is an important check against error in science and engineering, but is also useful in practical "dead-reckoning" estimates (e.g., figuring out whether 5 gallons of paint will cover the garage while you are standing in the aisle as WalMart).

I thought people here might get a kick out of trying their mental math skills on today's worksheet. The students have memorized the following facts using flashcards; these are sufficient for rough but reasonable estimates of the expressions on the worksheet. None of the problems should take more than a minute, and no more than a few strokes of a pencil (of course, the more accuracy you need, the more time will be required):

The squares of integers from 1 through 20.
The square roots of 2, 3, and 5 to three decimal places.
Tricks for factoring integers.
The powers of 2 from exponent 0 through exponent 10; and powers of 3 from exponent 0 through exponent 6.

For example, the following hints are helpful:

For problem number 1: 18 * 18 = 324
For problem number 10: 2^10 = 1,024


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1. sqr(3.24)
2. sqr(1.94)
3. What is the rightmost digit of 3^196?
4. fourth root of 3.62
5. sqr(32412)
6. sqr(300)
7. cube root of ((3^12)*(5^6))
8. 11 times 1357986421
9. sqr(432)
10. 8^100

[Latro]

Some of these are very blunt estimates:

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sqrt(3.24) = 0.1*sqrt(324)=0.1*18=1.8
sqrt(1.94) = about sqrt(1.96)=0.1*sqrt(196)=1.4.
multiplicative group of 3 mod 10: 3 9 7 1, 196 is evenly divisible by 4, answer is 1.
4throot(3.62) = about 1+2.62/4 = about 1+2.4/4 = 1.6. The estimate went down because the original is known to be an overshot, still didn't go low enough. A pretty good estimate averages 1, which is too small, and 1.6, which is too big, to give 1.3.
sqrt(32412) = about sqrt(32400) = sqrt(324)*sqrt(100)=180. This is too small; a better but too high estimate is 180+12/360 = 180+1/30.
sqrt(300) = about 17+11/34 = about 17+1/3, this should be a bit too big.
cube root(3^12 5^6) = 3^4 5^2 = 225*9 = 1800+225=2025
11*1357986421 = about 1360000000+1360000000 = 1496000000
sqrt(432) = about 21 - 9/42, should be too big. 20+32/40=20.8 will also be close but too big.
8^100 = (2^3)^100 = 2^300 = about (10^3)^30 = 10^90

Linear approximations are pretty awesome for roots if you know a fair number of perfect squares/cubes/etc. How do you suggest doing the fourth root one efficiently, though? It's a fair ways away from any perfect power of four, so it's not easy to get a decent estimate, especially since exploiting the "decimal point shift" trick for fourth roots of numbers less than 10 requires knowing fourth powers in the tens of thousands, which is not easy.

[Monte314]

Yeah, the fourth is a bit tougher. I suggested they think about it like this:

(3.62)^(1/4) = sqr(sqr(3.62))~sqr(sqr(361/100))~sqr(sqr(19/10))=sqr(1.9)~sqr(196/100) = 14/10 = 1.4