Question

Simplify (−5p^3)^3, (−w^3/6)^−2, (1/2r^6)^-6, (4s^5t^−7/−2s^−2t^4)^3, and (3x^3y^0x*−2)^4⋅(y^2x^−4^5xy^−8)^3.

Answer 1

$Use:\\\\(ab)^n=a^nb^n\\\\\left(\dfrac{a}{b}\right)^n=\dfrac{a^n}{b^n}\\\\(a^n)^m=a^{nm}\\\\\dfrac{a^n}{a^m}=a^{n-m}\\\\a^n\cdot a^m=a^{n+m}\\\\a^{-n}=\dfrac{1}{a^n}\\--------------------------------$  

$(-5p^3)^3=(-5)^3(p^3)^3=-125p^{(3)(3)}=-125p^9\\\\\left(-\dfrac{w^3}{6}\right)^{-2}\right)=\left(-\dfrac{6}{w^3}\right)^2=\left(\dfrac{6}{w^3}\right)^2=\dfrac{6^2}{(w^3)^2}=\dfrac{36}{w^{(3)(2)}}=\dfrac{36}{w^6}\\\\\left(\dfrac{1}{2r^6}\right)^{-6}=(2r^6)^6=2^6(r^6)^6=64r^{(6)(6)}=64r^{36}\\\\\left(\dfrac{4s^5t^{-7}}{-2s^{-2}t^4}\right)^3=\left(-\dfrac{4}{2}s^{5-(-2)}t^{-7-4}\right)^3=\left(-2s^7t^{-11}\right)^3\\\\=(-2)^3(s^7)^3(t^{-11})^3=-8s^{21}t^{-33}$

$(3x^3y^0x^{-2})^4(y^2x^{-4^5}xy^{-8})^3=(3x^{3+(-2)}1)^4(y^{2+(-8)}x^{-1024+1})^3.$

$=(3x^1)^4(y^{-6}x^{-1023})^3=3^4x^4(y^{-6})^3(x^{-1023})^3=81x^4y^{-18}x^{-3069}\\\\=81x^{4+(-3069)}y^{-18}=81x^{-3065}y^{-18}$