Simplify `(a^(4)b^(5//3))^(-3//4)`.

Answer

To simplify the expression \((a^{4}b^{\frac{5}{3}})^{-\frac{3}{4}}\), we will follow these steps:

### Step 1: Apply the Power of a Product Rule
Using the rule \((xy)^m = x^m y^m\), we can split the expression:
\[ (a^{4}b^{\frac{5}{3}})^{-\frac{3}{4}} = a^{4 \cdot -\frac{3}{4}} \cdot b^{\frac{5}{3} \cdot -\frac{3}{4}} \]

### Step 2: Simplify the Exponents
Now we will calculate the exponents:
1. For \(a\):
\[ 4 \cdot -\frac{3}{4} = -3 \]
2. For \(b\):
\[ \frac{5}{3} \cdot -\frac{3}{4} = -\frac{5}{4} \]

So, we have:
\[ a^{-3} \cdot b^{-\frac{5}{4}} \]

### Step 3: Rewrite with Positive Exponents
To express the negative exponents as positive, we use the property \(x^{-n} = \frac{1}{x^n}\):
\[ a^{-3} = \frac{1}{a^{3}} \quad \text{and} \quad b^{-\frac{5}{4}} = \frac{1}{b^{\frac{5}{4}}} \]

Thus, we can rewrite the expression as:
\[ \frac{1}{a^{3}b^{\frac{5}{4}}} \]

### Final Answer
The simplified form of the expression \((a^{4}b^{\frac{5}{3}})^{-\frac{3}{4}}\) is:
\[ \frac{1}{a^{3}b^{\frac{5}{4}}} \]

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