Powers Roots
Square a number
The product of a number times itself is the square of that number. So 3
squared is 3 times 3 or 9.
The calculator has a squared key. It is quite useful to minimize data entry
for difficult numbers.
Square Root of a number
The square root of a number is that value which when multiplied by itself
produces the number, itself. So the square root of 9 would be what number when
multiplied by itself would yield 9? Well in your head you can figure it to be 3.
3 times 3 equals 9. So the square root of 9 is 3.
The calculator has a square root key which makes the process painless to
find the square root of a number.
Note: When there are two factors under the radical (square root sign) then
you must first perform what ever math operation is identified for the factors,
like division or multiplication. Then press the equals sign. This will yield the
result of what is to be done with the factors.
Lastly, press the square root key to find the square root of the quantity
under consideration.
Find practice problems HERE.
Find audio support HERE.
Extras For Experts
In the department of more than you ever wanted to know, here is the technique
of solution to find the square root of a number without a calculator, as if
anyone would ever want to do that.
square root
(algorithm)
Definition: This describes a "long hand" or manual
method of calculating or extracting square roots. Calculation of a square root
by hand is a little like long-hand division.
Suppose you need to find the square root of 66564. Set up a
"division" with the number under the radical. Mark off pairs of
digits, starting from the decimal point. (Here the decimal point is a period (.)
and a comma (,) marks pairs of digits.)
___________
\/ 6,65,64.
Look at the leftmost digit(s) (6 in this case). What is the largest number whose
square is less than or equal to it? It is 2, whose square is 4. Write 2 above,
write the square below and subtract.
__2________
\/ 6,65,64.
-4
----
2
Now bring down the next two digits (65). The next "divisor" is double
the number on top (2x2=4) and some other digit in the units position (4_).
__2________
\/ 6,65,64.
-4
-----
4_ )
265
What is the largest number that we can put in the units and multiply times the
divisor and still be less than or equal to what we have? (Algebraically, what is
d such that d × 4d ≤ 265?) It looks like 6 might work (since 6 * 40 =
240), but 6 is too big, since 6 * 46 = 276.
__2__6_____
\/ 6,65,64.
-4
-----
46 )
265
276 TOO BIG
So try 5 instead.
__2__5_____
\/ 6,65,64.
-4
-----
45 )
265
-225
-------
40
Repeat: bring down the next two digits, and double the number on top (2x25=50)
to make a "divisor", with another unit.
__2__5_____
\/ 6,65,64.
-4
-----
45 )
265
-225
-------
50_ )
4064
It looks like 8 would work. Let's see.
__2__5__8__
\/ 6,65,64.
-4
-----
45 )
265
-225
-------
508 )
4064
-4064
------
0
So the square root of 66564 is 258. You can continue for as many decimal
places as you need: just bring down more pairs of zeros.
Why does this work?
Consider (10A + B)² = 100A² + 2 × 10AB + B² and think about finding the
area of a square. Remember that 10A + B is just the numeral with B in the units
place and A in the higher position. For 42, A=4 and B=2, so 10 × 4 + 2 = 42.
The area of the two skinny rectangles is 2 × 10A × B. The tiny square is B².
If we know A and the area of the square, S, what B should we choose?
We previously subtracted A² from S. To scale to 100A², we bring down two
more digits (a factor of 100) of the size of S. We write down twice A (2A), but
shifted one place to leave room for B (10 × 2A or 2 × 10A). Now we add B to
get 2 × 10A + B. Multiplying by B gives us 2 × 10AB + B². When we subtract
that from the remainder (remember we already subtracted A²), we have subtracted
exactly (10A + B)². That is, we have improved our knowledge of the square root
by one digit, B.
We take whatever remains, scale again by 100, by bringing down two more
digits, and repeat the process.
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