Question

`(-27a^9)^(-2/3)` = ______.

Options

• `(-9)/a^6`

• `1/(9a^6)`

• `3a^6`

• `-27a^-6`

Answer 1

`(-27a^9)^(-2/3)` = `underlinebb(1/(9a^6))`.

Explanation:

We are asked to simplify the expression:

`(-27a^9)^(-2/3)`

Step 1: Break down the expression

We will apply the exponent rules step by step.

The expression has two parts: –27 and a⁹. Let’s first handle these separately.

Step 2: Simplify `(-27)^(-2/3)`

We know that –27 = (–3)³.

So, `(-27)^(-2/3) = [(-3)^3]^(-2/3)`

= `(-3)^(3 xx - 2/3)`

= (–3)^–2

Now, apply the negative exponent rule `a^-n = 1/a^n`:

`(-3)^-2 = 1/(-3)^2 = 1/9`

Step 3: Simplify `(a^9)^(-2/3)`

Using the exponent rule (a^m)ⁿ = a^{m × n}, we have:

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

Step 4: Combine both parts

Now, putting the simplified parts together:

`(-27a^9)^(-2/3) = 1/9 xx a^-6 = 1/(9a^6)`

The simplified expression is `1/(9a^6)`.