How do you simplify #\frac { - 18p ^ { - 2} r ^ { 3} h ^ { - 8} v ^ { - 8} } { 9a ^ { - 1} p ^ { 3} r ^ { 3} v ^ { - 2} w }#?

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Answer 1 · 2018-04-17T21:19:35

#= -(2a)/(p^5h^8v^6w)#

#\frac { - 18p ^ { - 2} r ^ { 3} h ^ { - 8} v ^ { - 8} } { 9a ^ { - 1} p ^ { 3} r ^ { 3} v ^ { - 2} w }#

First, let's separate the like-terms to make it easier to see what's going on here:

#=(-18)/9 * 1/(a^(-1)) * p^-2/p^3 * r^3/r^3 * h^-8/1 * v^-8/v^-2 * 1/w#

#=(-18)/9 * a^0/(a^(-1)) * p^-2/p^3 * r^3/r^3 * h^-8/h^0 * v^-8/v^-2 * w^0/w#

since anything to the #0#th power is #1#.

The rules for like-terms and powers is:

We will need to use the latter.

#=-2 * a^(0-(-1)) * p^(-2-3) * r^(3-3) * h^(-8-0) * v^(-8-(-2)) * w^(0-1)#

#=-2 * a^(1) * p^(-5) * r^(0) * h^(-8) * v^(-6) * w^(-1)#

#=-2ap^-5h^-8v^-6w^-1#

Rewriting with only positive powers (if you would like to):

#= -(2a)/(p^5h^8v^6w)#

Answer 2 · 2018-04-17T21:24:39

#-(2a)/(p^5h^8v^6w)# or #-2ap^-5h^-8v^-6w^-1#

first, you can find individual quotients:

#18/9 = -2#

#1/(a^-1) = a^1 = a#

#p^-2/p^2 = p^(-2 - 3) = p^-5#

#r^3/r^3 = r^(3-3) = r^0 = 1#

#h^-8/1 = h^-8#

#v^-8/v^-2 = v^(-8 - -2) = v^(-8 + 2) = v^-6#

#1/w = w^-1#

multiplying all of these together gives a simplified version of the fraction.

hence, the expression can be written as #-2ap^-5h^-8v^-6w^-1#.

if you want to have the numerator as only positive exponents, you can write the negative exponents in the previous expression in the denominator.

this gives #-(2a)/(p^5h^8v^6w)#.