May 7, 2015 · $\log_2 (3) \approx 1.58496$ as you can easily verify. $ (\log_2 (3))^2 \approx (1.58496)^2 \approx 2.51211$. $2 \log_2 (3) \approx 2 \cdot 1.58496 \approx 3.16992$. $2^ {\log_2 …

Nov 17, 2013 · That's because the $9$ on the right hand side could have come from squaring a $3$ or from squaring a $-3$. So, when you square both sides of an equation, you can get extraneous …

Jan 2, 2022 · we can square both side like this: $ x^2= 2$ But I don't understand why that it's okay to square both sides. What I learned is that adding, subtracting, multiplying, or dividing both sides by …

Apr 20, 2016 · I understand that you can't really square on both the sides like I did in the first step, however, if this is not the way to do it, then how can you really solve an equation like this one (in …

7 Short answer: We can't simply square both sides because that's exactly what we're trying to prove: $$0 < a < b \implies a^2 < b^2$$ More somewhat related details: I think it may be a common …

May 26, 2020 · I took a look at square root. Squaring the number means x^2. And if I understood the square root correctly it does a bit inverse of squaring a number and gets back the x. I had a friend tell …

If you want the negative square root, that would be $-\sqrt {4} = -2$. Both $-2$ and $2$ are square roots of $4$, but the notation $\sqrt {4}$ corresponds to only the positive square root.

Dec 5, 2025 · Thus the values approach $\pi/2$ from above, so $$\inf_ {\substack {x\in\mathbb {Z}\\x\ \text {not a square}}} (1+\sqrt {x})|\sin (\pi\sqrt {x})| =\frac {\pi} {2}.$$

A key feature of least squares (which a median-based approach lacks) is that it is unbiased, i.e., the sum of the errors is zero. By the Gauss-Markov Theorem, least-squares is the best linear unbiased …

HINT: You want that last expression to turn out to be $\big (1+2+\ldots+k+ (k+1)\big)^2$, so you want $ (k+1)^3$ to be equal to the difference $$\big (1+2+\ldots+k+ (k+1)\big)^2- (1+2+\ldots+k)^2\;.$$ …