MATH SOLVE

4 months ago

Q:
# Estimate square root of 640 to the nearest hundredths place show your work

Accepted Solution

A:

The assumption a priori is that calculators are not allowed.

Everything has to be calculated in the head.

What can help is to simplify the problem

640=2^7*5=10*(2^3)^2

so

sqrt(640)=8sqrt(10)

If you know the square-root of basic numbers, as most people do, then you know that sqrt(10)=3.162277..., or at least 3.162

and 8 times that is 25.2982...

or 25.30 (accurate to the hundredth).

On the other hand, we cannot expect everybody to memorize square-roots, so we can still work it out in the head.

We know that 25*25=625

(there is an easy way to work that out, if you would like to know, post a separate question and send me the question number. You can always memorize, but not necessary for this one).

Apply Newton's method, with x0=25, then

x1=25-(25^2-640)/(2*25)

=25+15/50

=25.3

Apply Newton's method again,

x2=25.3-(25.3^2-640)/(2*25.3)

[25.3^2 = 25^2+2*0.3*25+0.3^2=625+15+0.09=640.09, doable in the head]

=25.3-(640.09-640)/(2*25.3)

=25.3-0.09/50.6 (do 25.3-0.09/50, it's quite close)

=25.3-0.0018

=25.2982

=25.30 (to 2 decimal places)

exact value is 25.29822128

You can also find it to as many places as you wish, without calculator, but with pen and paper. The method is similar to division, but with an extra step or two.

Three "registers"A=square-root, best estimateB=640 and the remaindersC=multiplier=2*square-root

Take a rough square-root less than the real number. We take the initial approximation A=2525*25=625B=640-625=15C=50 =2A

To continue, we need to find a single digit decimal number multiplied by C+the digit will be less than B.Try 0.2

0.2*(C+0.2)=0.2*50.2=10.04 <15Update: A=25.2, B=15-10.04=4.96, C=2A=50.4

Now we need another digit to continuetry 0.090.09*50.49=4.5441 < 4.96Update: A=25.29, B=4.6-4.5441=0.4159, C=50.58

Continue to confirm 2nd digit (hundredth)try 0.0080.008*50.588=0.404704A=25.298B=0.4159-0.404704=0.011196C=50.596...

Since we have already to the thousandth place, we can stop and say that sqrt(640)=25.30 (to the hundreth place)

Everything has to be calculated in the head.

What can help is to simplify the problem

640=2^7*5=10*(2^3)^2

so

sqrt(640)=8sqrt(10)

If you know the square-root of basic numbers, as most people do, then you know that sqrt(10)=3.162277..., or at least 3.162

and 8 times that is 25.2982...

or 25.30 (accurate to the hundredth).

On the other hand, we cannot expect everybody to memorize square-roots, so we can still work it out in the head.

We know that 25*25=625

(there is an easy way to work that out, if you would like to know, post a separate question and send me the question number. You can always memorize, but not necessary for this one).

Apply Newton's method, with x0=25, then

x1=25-(25^2-640)/(2*25)

=25+15/50

=25.3

Apply Newton's method again,

x2=25.3-(25.3^2-640)/(2*25.3)

[25.3^2 = 25^2+2*0.3*25+0.3^2=625+15+0.09=640.09, doable in the head]

=25.3-(640.09-640)/(2*25.3)

=25.3-0.09/50.6 (do 25.3-0.09/50, it's quite close)

=25.3-0.0018

=25.2982

=25.30 (to 2 decimal places)

exact value is 25.29822128

You can also find it to as many places as you wish, without calculator, but with pen and paper. The method is similar to division, but with an extra step or two.

Three "registers"A=square-root, best estimateB=640 and the remaindersC=multiplier=2*square-root

Take a rough square-root less than the real number. We take the initial approximation A=2525*25=625B=640-625=15C=50 =2A

To continue, we need to find a single digit decimal number multiplied by C+the digit will be less than B.Try 0.2

0.2*(C+0.2)=0.2*50.2=10.04 <15Update: A=25.2, B=15-10.04=4.96, C=2A=50.4

Now we need another digit to continuetry 0.090.09*50.49=4.5441 < 4.96Update: A=25.29, B=4.6-4.5441=0.4159, C=50.58

Continue to confirm 2nd digit (hundredth)try 0.0080.008*50.588=0.404704A=25.298B=0.4159-0.404704=0.011196C=50.596...

Since we have already to the thousandth place, we can stop and say that sqrt(640)=25.30 (to the hundreth place)