Question

Simplify:

`((512a^6)/(729b^-9))^((-2)/3)`

Answer 1

We are asked to simplify:

`((512a^6)/(729b^-9))^(-2/3)`

Step 1: Recognize bases

• 512 = 8³ = 2⁹
• 729 = 9³ = 3⁶

So the expression becomes:

`((2^9a^6)/(3^6b^-9))^(-2/3)`

Step 2: Simplify inside

Since b^–9 is in denominator, it goes up:

`((2^9a^6b^9)/(3^6))^(-2/3)`

Step 3: Apply the exponent `-2/3`

Raise each factor:

= `(2^9)^(-2/3) * (a^6)^(-2/3) * (b^9)^(-2/3) * (3^6)^(2/3)`

Step 4: Simplify exponents

`(2^9)^(-2/3) = 2^(9* - 2/3) = 2^-6 = 1/2^6 = 1/64`

`(a^6)^(-2/3) = a^(6* - 2/3) = a^-4 = 1/a^4`

`(b^9)^(-2/3) = b^(9* -2/3) = b^-6 = 1/b^6`

`(3^6)^(2/3) = 3^(6 * 2/3) = 3^4 = 81`

Step 5: Combine

= `81/(64a^4b^6)`