# Tagged Questions

219 views

### Difference between $\sqrt{x^2}$ and $(\sqrt{x})^2$

According to my logic, $$\large\sqrt{x^2} = x^{2\times \frac{1}{2}} = x = x^{\frac{1}{2}\times 2}={(\sqrt{x})}^2$$ But when I look at the graphs of these guys, they're totally different. Edit: ...
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### Square root of a squared number changes sign, which to apply first?

Heres something Ive always found interesting. Supose we have a variable $x$, and $x$ equals a negative number: Say: $$x=-17$$ Now, I can apply a square to both sides of the equation and preserve ...
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### Is $\left((-1)^2\right)^\frac12 = (-1)^\left(2\cdot\frac12\right)$? [duplicate]

I'm feeling confused. If I square 1 and -1, the answers should be equal: $1^2 = (-1)^2$ Then I take both sides to the power of $\frac12$: $\left(1^2\right)^\frac12 = \left((-1)^2\right)^\frac12$ ...
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### How to show that all roots of $(11+v)q^3-18q^2+9q-2$ have their absolute value less than 1.

The equation is $(11+v)q^3-18q^2+9q-2=0$, where $v>0$ I need to show that either absolute value of all the roots is not greater than one or there exists a root $q: |q|>1$. Using Weierstrass ...
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### Ways to compute roots of complex numbers

I know how to use the De Moivre's Formula, but to caculate it one need to use caclulator. Is there any better way to take roots of complex numbers that is more "caclulator-free"? I am particulary ...
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### Why are the roots of the polynomial $z^N = a^N$ equal to $z_k = a \ e^{j\frac{2 \pi k}{N}}$?

I am trying to understand equation 3.28 from this image in my book. I get everything that the author is saying, except for when he finds the roots, (zeros), of $z^N = a^N$. Of course, there are ...
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### Rational exponents: prove some states

In some rational exponent expressions the solution isn't a real number why? Example (explain what I mean): \begin{align} \Big(-x\Big)^{1/n}=\left\{\text{is not a real number}\right\} \end{align} ...
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### Multiplying two complex numbers using only three multiplications of real numbers

I have problem given below. Show how two complex numbers $(a+ib)$ and $(c+id)$ may be multiplied using only three multiplications of real numbers, where $i=\sqrt{-1}$. You may use any number of ...
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### nth roots of negative numbers

Disclaimer: I know what complex numbers are. Let $x,\space n\in\Bbb R$ What is the complex algebraic solution to $\sqrt[n]{-x}$? Could I have a 'general' formula and a walk through on how to ...
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### Which roots of a negative number can be done?

I'm an Android programmer and am working on a graphing calculator. I have been looking for the limits on which roots can be done. I have a decent understanding of mathematics but can not seem to find ...
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### rearrange $z \mapsto z^2 + c$

Mathematics, some of its magic is that a lot is known about how to rearrange its statements (equations). Given the Mandelbrot Set: $z \mapsto z² + c$ (or more precisely) $z_{i+1} = z_i ^2 + c$ ...
I am confused about some exponent behavior. $$(-2)^{7.6} = (-2)^{\frac{76}{10}} = ((-2)^{76})^{\frac{1}{10}} = ((-2)^{\frac{1}{10}})^{76}$$ Is there something wrong in this logic? When I plug the ...