How do you simplify \frac { - 18p ^ { - 2} r ^ { 3} h ^ { - 8} v ^ { - 8} } { 9a ^ { - 1} p ^ { 3} r ^ { 3} v ^ { - 2} w }−18p−2r3h−8v−89a−1p3r3v−2w?

2 Answers
NJ
Apr 17, 2018

= -(2a)/(p^5h^8v^6w)=−2ap5h8v6w

Explanation:

\frac { - 18p ^ { - 2} r ^ { 3} h ^ { - 8} v ^ { - 8} } { 9a ^ { - 1} p ^ { 3} r ^ { 3} v ^ { - 2} w }−18p−2r3h−8v−89a−1p3r3v−2w

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=−189⋅1a−1⋅p−2p3⋅r3r3⋅h−81⋅v−8v−2⋅1w

=(-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=−189⋅a0a−1⋅p−2p3⋅r3r3⋅h−8h0⋅v−8v−2⋅w0w

since anything to the 00th power is 11.

The rules for like-terms and powers is:

  • Multiplication of terms requires addition of powers
  • Division of terms requires subtraction of powers

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⋅a0−(−1)⋅p−2−3⋅r3−3⋅h−8−0⋅v−8−(−2)⋅w0−1

=-2 * a^(1) * p^(-5) * r^(0) * h^(-8) * v^(-6) * w^(-1)=−2⋅a1⋅p−5⋅r0⋅h−8⋅v−6⋅w−1

=-2ap^-5h^-8v^-6w^-1=−2ap−5h−8v−6w−1

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

= -(2a)/(p^5h^8v^6w)=−2ap5h8v6w

LM
Apr 17, 2018

-(2a)/(p^5h^8v^6w)−2ap5h8v6w or -2ap^-5h^-8v^-6w^-1−2ap−5h−8v−6w−1

Explanation:

first, you can find individual quotients:

18/9 = -2189=−2

1/(a^-1) = a^1 = a1a−1=a1=a

p^-2/p^2 = p^(-2 - 3) = p^-5p−2p2=p−2−3=p−5

r^3/r^3 = r^(3-3) = r^0 = 1r3r3=r3−3=r0=1

h^-8/1 = h^-8h−81=h−8

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

1/w = w^-11w=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^-1−2ap−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)−2ap5h8v6w.

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