$$ \left. \begin{array} { l } { 8 a ^ { 6 } b ^ { - 2 } } \ { 27 a ^ { - 3 } b } \end{array} ight) ^ { - \frac { 2 } { 3 } } $$

Question

$$ \left. \begin{array} { l } { 8 a ^ { 6 } b ^ { - 2 } } \ { 27 a ^ { - 3 } b } \end{array} ight) ^ { - \frac { 2 } { 3 } } $$

UpStudy AI · Accepted Answer

To simplify the expression:

$$
\left( \frac{8a^{6}b^{-2}}{27a^{-3}b} \right)^{-\frac{2}{3}}
$$

**Step 1: Simplify the Fraction Inside the Parentheses**

$$
\frac{8a^{6}b^{-2}}{27a^{-3}b} = \frac{8}{27} \cdot a^{6 - (-3)} \cdot b^{-2 - 1} = \frac{8}{27} \cdot a^{9} \cdot b^{-3}
$$

So, the expression becomes:

$$
\left( \frac{8}{27}a^{9}b^{-3} \right)^{-\frac{2}{3}}
$$

**Step 2: Apply the Exponent to Each Factor**

$$
\left( \frac{8}{27} \right)^{-\frac{2}{3}} \cdot a^{9 \cdot -\frac{2}{3}} \cdot b^{-3 \cdot -\frac{2}{3}} = \left( \frac{8}{27} \right)^{-\frac{2}{3}} \cdot a^{-6} \cdot b^{2}
$$

**Step 3: Simplify $$\left( \frac{8}{27} \right)^{-\frac{2}{3}}$$**

$$
\left( \frac{8}{27} \right)^{-\frac{2}{3}} = \left( \frac{2^3}{3^3} \right)^{-\frac{2}{3}} = \left( \frac{2}{3} \right)^{-2} = \left( \frac{3}{2} \right)^{2} = \frac{9}{4}
$$

**Step 4: Combine All Parts**

$$
\frac{9}{4} \cdot a^{-6} \cdot b^{2} = \frac{9b^{2}}{4a^{6}}
$$

**Final Simplified Form:**

$$
\boxed{\dfrac{9 b^{2}}{4 a^{6}}}
$$
$$ \frac{9b^{2}}{4a^{6}} $$

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