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sqrt(18 a^6 b^3 c^5) = 3 a^3 b c^2 sqrt(2 b c)
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Proof / simplification:
sqrt(18 a^6 b^3 c^5)
= sqrt(9·2 · (a^3)^2 · b^2 · c^4 · (b c))
= sqrt(9)·sqrt((a^3)^2)·sqrt(b^2)·sqrt(c^4)·sqrt(2 b c)
= 3·|a^3|·|b|·c^2·sqrt(2 b c).
Since |a^3| = |a|^3, this is 3 |a|^3 |b| c^2 sqrt(2 b c). If you assume a ≥ 0 and b ≥ 0 (and of course b c ≥ 0 so the inner sqrt is real), the absolute values drop and you get
sqrt(18 a^6 b^3 c^5) = 3 a^3 b c^2 sqrt(2 b c).
sqrt(18 a^6 b^3 c^5)
= sqrt(9·2 · (a^3)^2 · b^2 · c^4 · (b c))
= sqrt(9)·sqrt((a^3)^2)·sqrt(b^2)·sqrt(c^4)·sqrt(2 b c)
= 3·|a^3|·|b|·c^2·sqrt(2 b c).
Since |a^3| = |a|^3, this is 3 |a|^3 |b| c^2 sqrt(2 b c). If you assume a ≥ 0 and b ≥ 0 (and of course b c ≥ 0 so the inner sqrt is real), the absolute values drop and you get
sqrt(18 a^6 b^3 c^5) = 3 a^3 b c^2 sqrt(2 b c).
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