How to solve this expression?  `(5/6)^(-3)[(5/6)^(3)-:(36/25)^(-2)]^(3){-[(-6/5)^(2)]^(-4)}^(3)= ?`Good morning, how do you solve this expression? thank you! 

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lemjay | High School Teacher | (Level 3) Senior Educator

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`(5/6)^(-3) [ (5/6)^3 -: (36/25)^(-2) ]^3 { -[(-6/5)^2]^(-4)}^3`

Simplify the expressions inside the braces, brackets and parenthesis. To do so, start from the innermost expression going outside.

Let's start to simplify the expression inside the braces.

>> `{ - [ (-6/5)^2]^-4 } ^3`

Since the power of (-6/5)^2 is an even number , the resulting sign is positive. And use the property of exponents which is  `(a^m/b^n)^k = a^(m*k)/b^(n*k)`  to simplify.

     ` { - [6^2/5^2]^-4}^3`

Again, use the property `(a^m/b^n)^k = a^(m*k)/b^(n*k)` .

   ` { - 6^(2(-4))/5^(2(-4)) }^3 = {-6^(-8)/5^(-8) }^3`

Since the outermost power is an odd number, the resulting sign of the expression inside the braces is negative and apply the same property of exponents.

   `-6^(-8*3)/5^(-8*3) = - 6^(-24)/5^(-24)`

So we have,

`=(5/6)^(-3)[(5/6)^3 -: (36/25)^(-2)]^3 *(-6^(-24)/5^(-24))`

Next, simplify the expression inside the brackets.

>> `[(5/6)^3 -: (36/25)^(-2)}^3`

Express 36 as `6^2` and 25 as `5^2` .

     `[ (5/6)^3 -: (6^2/5^2)^(-2) ]^3`

Apply the exponent property `(a^m/b^n)^k = a^(m*k)/b^(n*k)` .

     `[ 5^3/6^3 -:6^(2(-2))/5^(2(-2)) ]^3 =[ 5^3/6^3 -: 6^(-4)/5^(-4)]^3`

To divide fractions, flip the divisor and change the operation from division to multiplication.

     `[5^3/6^3 * 5^(-4)/(6^(-4))]^3 = [(5^3*5^(-4))/(6^3*6^(-4))]^3`

To multiply same base, add the exponents.

    `[5^(3+(-4))/6^(3+(-4)) ]^3 = [5^(-1)/6^(-1)]^3 = 5^(-3)/6^(-3)`

 So we have,

`= (5/6)^(-3) *5^(-3) / 6^(-3)* (-6^(-24)/5^(-24)) = 5^(-3)/6^(-3) * 5^(-3)/6^(-3) * (- 6^(-24)/5^(-24))`

`= -(5^(-3) * 5^(-3) * 6^(-24))/(6^(-3)*6^(-3)*5^(-24)) = -(5^((-3)+(-3)) * 6^(-24))/(6^((-3)+(-3))*5^(-24)) =-(5^(-6)*6^(-24))/(6^(-6)*5^(-24))`

To divide same base, subtract the exponents `(a^m/a^n = a^(m-n))` .

`=-5^(-6-(-24)) 6^((-24)-(-6)) =- 5^18*6^(-18)`

To express 6 with positive exponent, apply the negative exponent rule which is `a^(-m)= 1/a^m` .

`= -5^18/6^18`

Hence, `(5/6)^(-3) [ (5/6)^3 -: (36/25)^(-2) ]^3 { -[(-6/5)^2]^(-4)}^3 = -5^(18)/6^(18)` .

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