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Simplifying various algebraic expressions

Please simplify the attached expressions. Thanks

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Notation:
^ means to the power of
sqrt(.) means square root of (.)
1|1/2 means 1 and a half, i.e. 3/2

Some of the algebraic properties that I will use:
i) a^(b*c) = (a^b)^c , e.g. a^8 = (a^4)^2 = (a^2)^4 (they are interchangeable)
ii) n-th root of (a) = a^(1/n), e.g. 3-rd root of 8 = 8^(1/3) = 2
iii) 1/a = a^(-1), e.g. 1/(x^4) = (x^4)^(-1) = x^(-4)

2a) sqrt(64 x^8 y^12) = sqrt( 8^2 (x^4)^2 (y^6)^2) by property i)
= 8 x^4 y^6 by property ii)

2b) 3rd-root(2^3 (x^2)^3 (y^5)^3) = 2 x^2 y^5 using properties i) and ii)

3a) ...

Solution Summary

This solutions shows how to simplify some kinds of algebraic expressions. The simplifying rules are explained in the beginning and the solution itself shows how they are applied to specific problems.

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