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How to Calculate Square Roots by Long Division
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Jun 28, 2025
In this video, I’ll demonstrate how to calculate the square root of 2 using the long division method, aiming for an accuracy of up to 0.001. Follow along step by step as we break down the process to achieve a precise result. Whether you’re learning this technique for the first time or refreshing your skills, this tutorial will guide you through the entire method.
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0:00
hello welcome to we learn daily in this
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lesson let's review how to calculate the
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square root using long division we'll
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find the square root of two accurate to
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three decimal places let's
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start first let's rewrite the square
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root of two next we need to break the
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number from which we're extracting the
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square root into groups of two digits
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each starting from the
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right if the number has an odd number of
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digits like in this case where the
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number consists of one digit we add a
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zero to the left of the number so we add
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a zero before the two now we need to
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extract the square root of the first
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group which is the largest whole number
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that doesn't exceed two the largest such
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number is one so we write down one as
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the whole number then Square this number
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and write the result underneath the
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group 1 squ is 1 so we subtract 1 from 2
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giving us a remainder of one next draw a
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division line and bring down the next
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two digits which are zeros making the
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remainder 100 since we've brought down
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digits after the decimal point don't
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forget to place a decimal point in the
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quotient as well to continue double the
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number we've written in the quotient so
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far which is one this product is two
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write two to the left of a blank space
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under the line
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now we need to find a digit to place in
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the blank space so that when this new
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two-digit number is multiplied by this
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same digit the product is as close to
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100 as possible without exceeding it
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suppose we try five 5 * 25 is 125 which
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exceeds 100 so it's too large let's try
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four 4 * 24 gives us 96 Which is less
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than 100 we write down four in the
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quotient
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now subtract 96 from 100 to get a
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remainder of
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four bring down the next two digits
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which are zeros making the new number
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400 next Double the current quotient
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which is 14 and write it to the left of
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a blank
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space this product is 28 Now find the
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next digit if we try one the product is
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281
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Which is less than
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400 write down one in the quotient and
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subtract 281 from 400 leaving a
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remainder of 119 we now bring down the
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next two digits and we continue the same
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process we need to draw another division
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line and multiply the current number
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ignoring the decimal point we multiply
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by two and get
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282 we write down 282 and place a DOT
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next to it then we place another another
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dot under it and underline the
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result now we need to choose a digit so
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that when this number is multiplied by
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The Chosen digit the product is as large
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as possible without exceeding
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11,900 we choose four and multiply it by
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2,824 getting
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11,296 we subtract and get a remainder
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of four here we have a zero and since we
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borrowed earlier we are left with six
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the digit that replaces the dot will be
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four since we need the result accurate
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to the nearest thousandth we continue
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dividing until we reach
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10,000s so that we can round the result
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later next we bring down two more digits
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to zeros and multiply the entire current
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number by two again we always multiply
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by two the result is
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2,828 we leave space for the dot again
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and proceed to find the appropriate
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digit let's try through
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if we multiply by 3 the result exceeds
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60,000 so let's try two we multiply 2x
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28282 and get
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5656 4 now that we've gone through this
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process we can round the square root of
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2 to the nearest thth the square root of
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2 is approximately
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1.414 but less than 1.
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1415 therefore we can write the final
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result as approximately 1
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414 the problem is solved if you
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understood the solution don't forget to
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like the video and leave a comment thank
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you for watching and I will see you in
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the next video goodbye
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