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KEYMACRO is an extension for the Apple Wireless Keyboard for use in Mac OS X. Simply download the application, and it will automatically detect your Apple wireless keyboard, and offer you the option to save a macro of the keyboard’s keys into your.Mac account. KEYMACRO also offers the user the option to save a shortcut with a different key than a standard keyboard macro, like for instance, the ¢ key for the return key. You can also store additional symbols, as well as Macros. Once the.Mac account is synchronized, the Macro’s content will also be uploaded to the Apple Wireless Keyboard. You can also save Macros directly to your Macbook’s Home directory.
CyberLink’s PowerDVD is a media player and suite developed by CyberLink Corporation. The software is mainly targeted at Windows OS users, but there are also versions for Mac OS X and Linux.
PowerDVD includes some applications that can be used to manage a wide range of multimedia content, including movies, music, TV shows, photographs, games and e-books. The suite comes with a DVD player and a set of tools to help you organize and watch your movies.
There are two main parts of the software: PowerDVD Player and PowerDVD Gallery. The former is a player that helps you manage your movies, while the latter is a gallery where you can put up all of your content.
PowerDVD Player is a media player that also comes with some advanced features, such as playback of all types of DVDs (not only the ones on a disc) with a single click, playback of 4K Ultra HD Blu-ray discs, an enhanced 3D mode, enhanced subtitle support and an option to play the DVD with surround sound, while a separate application is used to create 2D to 3D conversion files. The software also includes image filters, video effects and transitions.
PowerDVD Player can be enhanced with the PowerDVD Premium. The Premium includes all the functions of the regular version, plus the ability to play and convert in any format, plus a virtual disc library, Dolby Atmos Audio and Blu-ray 3D support.
PowerDVD Gallery is a digital media gallery and social media client for Windows that integrates directly into Windows Explorer, making it easy to access and manage your media in a single application.
It offers the basic features such as playing, burning and streaming movies, photos, music, e-books and other multimedia files, while also adding advanced options, including dynamic thumbnails, 384a16bd22

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Proof that $a^n=b^n$

I have a question about how to prove that $a^n=b^n$ for all $a,b \in \mathbb{Q}$ and $n \in \mathbb{N}$.
I know that this is true, because all algebraic numbers are algebraic integers and algebraic integers are closed under multiplication. But how would one go about proving this algebraically?
I would appreciate any help.


We have
a^n=b^n\iff (a^{ -1})^nb^n=(b^{ -1})^na^n=b^{ -n}(b^{n})^a=b^{ -n}b^n=(b^{ -1})^nb^n\iff b^{ -n}=a^{ -n}
where we used the fact that for $\alpha \in \Bbb Q$ the conjugate of $\alpha$ is $\alpha^{ -1}$.
Hence we have that
Let $p$ be a prime factor of $a^n-b^n$. Then we have
But $\sqrt[n]{a^n}-\sqrt[n]{b^n}=a^{\frac{n}{p}}-b^{\frac{n}{p}}$
and $\gcd(a,b) \mid a^{\frac{n}{p}}-b^{\frac{n}{p}}$
So either $p$ divides both of $a^{\frac{n}{p}}-b^{\frac{n}{p}}$ and $\sqrt[n]{a^n}-\sqrt[n]{b^n}$ or $p$ divides $\sqrt[n]{a^n}-\sq


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