algrebra2 ????

Question

ok how do u know the decimal of a radical(not the exact amount) like radical 3 without using a calculator

Answer 1

Are you referring to how to compute the square root?

If so, it's best done by a calculator. However, you don't need anything but the basic arithmetic operations to compute it.

If you have a calculator that can only add, subtract, multiply and divide you could use the following method to quickly compute a square root and it's much more efficient than continuous trial and error. The only guess work involved is in the 1st approximation and that part is easy.

You can do this without a calculator, but it would take some time. The advantage of this method is that it's efficient and uses minimal guesswork.

A quick example for the square root of 3:

1² = 1
2² = 4

Therefore, the square root of 3 is between 1 and 2.
Let's try 1.7 as the 1st approximation = a

1.7² = 2.89, not quite 3, but better than 1 or 2.

To get a better approximation:

b = (1.7 + (3 / 1.7)) / 2 = 1.7323529412

Let's keep only the 1st 3 decimals of this 2nd approximation.

1.732² = 2.999824

so 1.732 is a better approximation than the original 1.7

Repeat the process above replacing 1.7 with 1.732 and we get

b = (1.732 + (3 / 1.732)) / 2 = 1.7320508083

then

1.7320508083² = 3.0000000026

So in only 3 approximations we get 1.7320508083 as the square root of 3, correct to 9 figures.

If we used 1.7320508083 as the next approximation, the result would be even closer.

This method is mostly adapted for computer languages that do not have a square root function and allows for square roots to be easily be computed from the four basic arithmetic operations.


This method is outlined in detail below:







To find the square root of a positive number from scratch
by successively better approximations:

Let:

x = Number for which we desire the square root

a = Current approximation to square root of (x)

b = Next better approximation derived from current (a) value.


Given 1st approximation (a), to get the next better approximation (b) to the square root of (x) :

b = (a + (x / a)) / 2


If (b) is not precise enough, then substitute the current (b) value for (a) in the formula and repeat. Each cycle gives a better approximation until you grind it out to whatever degree of precision you require.




Example:

To find the square root of 19.613

We know that 4² = 16 and that 5² = 25

Therefore the value we seek is between 4 and 5

In this case, since 4² is closer to 19.613, let a=4

Thus, using 4 as a 1st approximation to SqrRoot(19.613)
in the formula gives:

x = 19.613

a = 4

b = (4 + (19.613 / 4)) / 2 = 4.451625

So, b=4.451625 becomes our 2nd approximation and is better than the 4 we started with initially.

Now, b² = 19.8169651406

which is closer to the square root of 19.613 than was the original a=4



Now let

a = 4.451625 = current (b) value

and compute a new value of (b) from

b = (4.451625 + (19.613 / 4.451625)) / 2 = 4.42871593414

b² = 19.6135248253, which is now accurate to 5 significant figures.



Let's do another cycle for an even better approximation:

The new a = current b = 4.42871593414

So,

the new b = (4.42871593414 + (19.613 / 4.42871593414)) / 2 = 4.4286566816

Now

b² = 19.6130000035

So, the 3rd approximation gives an error of only 0.0000000035 when squared, but we can still do better.



A 4th approximation based on the new (a) value finally gives:

b = (4.4286566816 + (19.613 / 4.4286566816)) / 2 = 4.42865668121

and b² = 19.613 to sufficient accuracy.

Therefore the square root of 19.613 is 4.42865668121 and it took only 4 iterations starting with 4 as the 1st approximation.


Using this method, you can grind out a square root to whatever level of precision is required within the ability of the computer or calculator being used.

You simply take the answer (b) from the previous cycle and plug it in as the new (a) value for the next cycle to zero in on the square root value.

This process iterates quickly by starting with the closest integer approximation to the required square root, as in this case starting with 4.


A similar process can be used to grind out cube roots, 4th roots, 5th roots, etc.




So, to summarize:

Given 1st approximation (a), the next better approximation (b) to the square root of (x) is:

b = (a + (x / a)) / 2

After each cycle, take the resulting (b) value and substitute it as the new (a) in the formula and repeat until desired precision is reached.

The final (b) value will be the desired square root of (x).

This method is also very easy to program in just about any computer language that can do floating point arithmetic.

Of course, (x) must be a non-negative value.

Answer 2

There is a way, that I learned a long time ago but have forgotten. You separate your number in blocks of two digits, then work on it sort of like long division -- but I forgot the process. Suppose you want the square root of 27098.326. Then you'd break it up to 2 70 98. 32 60 and work on it from there. (That answer is a 3-digit number, plus two decimals, beginning with 1.)

Another method is to take half the log, then find the inverse log. You can do that with tables. For example (not using a calculator), to get the square root of 27098 using a five-place printed log table, I have

log 27090 = 4.43281
log 27100 = 4.43297
Difference = 0.00016

Interpolate: add 90% of the difference to the lower number, or subtract 10% of the difference from the higher number:

4.43281 + .9 x .00016 = 4.432954
4.43297 - .1 x .00016 = 4.432954

Divide by 2 to get log x = 2.216477.

Look up the inverse log in a table. I see

log 164.60 = 2.216430
log 164.70 = 2.216690

The difference here is 0.000260. Now interpolate:

(log x - 2.216430)/.000260 = .000047/.000260 = 47/260 = 0.18 approximately. So our approximation for the square root of 27098 is 164.618.

Using a calculator, I see that the square root of 27098 is 164.6147. Our approximation using log tables was pretty good.

Answer 3

One way's by guess and check. Another's with an algorithm. There are others, to be certain, but the first two are explained at
http://www.homeschoolmath.net/teaching/square-root-algorithm.php

Answer 4

You don't. You guesstimate, or slap it into a calc. You SHOULD however, know the answer to things like sqrt 9 or cubrt 27.

Answer 5

come on why do you want to know, you are allowed to use a calculator in algebra 2.