How do you solve #x^ (-2/3) = 9#?

Answer 1
raise both sides to the #-3/2# power.
Then you have: #x^((-2/3)(-3/2))=9^(-3/2)#
#x=9^(-3/2)=(3^2)^(-3/2)#
#x= 3^-3#
#x=1/27#
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Answer 2

A trick using logs.

#x=1/27#

Given: #x^(-2/3)=9#

Take logs of both sides:

#ln(x^(-2/3))#=ln(9)#
#-2/3ln(x)=ln(9)#
#ln(x)=-3/2ln(9)#
#x=ln^(-1)[-3/2ln(9)]#
#x= 0.037color(white)(.)bar(037)# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ #1000x=37.037bar(037)# #ul(color(white)(1000)x=color(white)(0)0.037bar(037)larr" Subtract")# # color(white)("d") 999x=37#
#x=37/999 = 1/27# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ #color(blue)("Check")#
#x^(-2/3) = 1/(x^(2/3))#
Consider #x^(2/3)->(1/27)^(2/3)= 0.11bar1#
#1/(x^(2/3))=1/(0.11bar1) = 9 color(red)(larr" As required")#
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Answer 3

To solve ( x^{-\frac{2}{3}} = 9 ), raise both sides of the equation to the power of (-\frac{3}{2}) to get ( x = 9^{-\frac{3}{2}} ). Then, simplify the right side to find ( x ).

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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