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How do you find the square root of 50?

Answer 1

Noah Archer

#sqrt50=5sqrt2#

#sqrt50#

= #sqrt(2xxul(5xx5))#

= #5sqrt2#

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Answer 2

Cameron Arnold

#sqrt(50)# can be simplified as #5sqrt(2)#

We can also find rational approximations to it.

For example:

#sqrt(50) ~~ 7 14/197 ~~ 7.071066#

The square root of #50# is not a whole number, or even a rational number. It is an irrational number, but you can simplify it or find rational approximations for it.

First note that

#50 = 2 xx 5 xx 5#

contains a square factor #5^2#. We can use this to simplify the square root:

#sqrt(50) = sqrt(5^2*2) = sqrt(5^2)*sqrt(2) = 5 sqrt(2)#

Apart from simplifying it algebraically, what is its numerical value?

Note that #7^2 = 49#, so #sqrt(49) = 7# and #sqrt(50)# will be slightly larger than #7#.

In fact, since #50=7^2+1#, the square root of #50# is expressible as a very regular continued fraction:

#sqrt(50) = 7+1/(14+1/(14+1/(14+1/(14+1/(14+1/(14+...))))))#

This can be written as #sqrt(50) = [7;bar(14)]# where the bar over the #14# indicates the repeating part of the continued fraction.

We can terminate the continued fraction early to give us rational approximations for #sqrt(50)#.

For example:

#sqrt(50) ~~ [7;14] = 7+1/14 = 7.0bar(714285)#

#sqrt(50) ~~ [7;14,14] = 7+1/(14+1/14) = 7+14/197 ~~ 7.071066#

In fact:

#sqrt(50) ~~ 7.071067811865475244#

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Answer 3

Julian Avila

To find the square root of 50, you can use various methods such as:

Using a calculator or computer: Simply input "sqrt(50)" into a calculator or use the square root function in a computer program, which will give you the approximate value of the square root of 50.
Using prime factorization: Write 50 as a product of its prime factors (2 x 5 x 5), then take the square root of each prime factor. Simplify if possible.
Using estimation: Since 50 is between the squares of 7 and 8 (which are 49 and 64, respectively), you can estimate that the square root of 50 is between 7 and 8.
Using a numerical method such as the Babylonian method: This iterative algorithm can provide an approximation of the square root of a number. Start with an initial guess, then iteratively refine the guess until it converges to the square root of 50.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.