2015年5月12日 · Since it's continuous, the square root of any positive real number is always a well-defined positive real number: Given the positive gap-between-rationals which you want to take the square root of, the square root is the positive gap-between-rationals such that any rational greater than the square-root-gap squares to a rational greater than the ...

The general guideline of writing the square root as a fractional power and then using the power and chain rule appropriately should be fine however. Also, remember that you can simply pull out a constant when dealing with derivatives - see below.

Now, let's explore how to translate a square root function vertically. y = #sqrt(x) + 3# or y = #sqrt(x) - 4#. The addition or subtraction on the OUTSIDE of the square root function will cause the graph to translate up or down. Adding 3 will raise the graph up, and subtracting 4 will lower the graph by 4 units. (see graph)

$\begingroup$ Minor point: I notice quite a few elementary algebra books as well as some writers here taking the view that the n-th root of x is defined as x to the power 1/n. I disagree strongly. I disagree strongly.

2014年6月24日 · The square root of the square root of x is therefore $$\sqrt{\sqrt x} = (\sqrt x)^{1/2} = (x^{1/2})^{1/2} ...

Law of Exponents $\,\Rightarrow\,x = a^{1/2}\,$ is a positive root of $\, x^2 - a,\ $ assuming $\ a > 0.$ Further, by definition, $\,\sqrt{a}\ $ is also a positive root of $\, x^2 - a.$ If $\ a^{1/2}\neq \sqrt{a}\ $ then the quadratic $\,x^2-a\,$ has $> 2$ roots: $\,a^{1/2},\ {\pm}\sqrt{a},\,$ contradicting the theorem that a nonzero polynomial ...

2015年5月12日 · The suaqre root of a (non-negative) real number is non-negative by definition, but is there a similar decision for "the" square root of other (complex) numbers? $\endgroup$ – Wolfgang Kais Commented Jul 8, 2023 at 17:59

You are referring to the principal square root. From Wikipedia: "Although the principal square root of a positive number is only one of its two square roots, the designation 'the square root' is often used to refer to the principal square root."

I have a question regarding taking square roots in inequalities. I have a problem asking: Suppose $3x^2+bx+7>0$ for every real number x. Show that $|b|<2\\sqrt{21}$. In an earlier question i...

2015年9月16日 · It is beneficial to be able to estimate, or calculate exactly the square root of any number when a calculator is not present. For this example, we can start an estimation limit high and low as follows: #20xx20=400# is too low #30xx30=900# is too high, but closer to #784# If we raise the lower limit to #25#: #25xx25=625# is still too low, but ...