Questions: (a^3 / -3b^5)^4

Solution Steps

To simplify the given expression, we need to apply the power of a quotient rule, which states that \((\frac{a}{b})^n = \frac{a^n}{b^n}\). We will raise both the numerator and the denominator to the power of 4. Then, simplify the expression by applying the power rule for exponents, which states that \((a^m)^n = a^{m \cdot n}\).

We start with the expression \[ \left(\frac{a^{3}}{-3 b^{5}}\right)^{4}. \] Using the power of a quotient rule, we can rewrite this as \[ \frac{(a^{3})^{4}}{(-3 b^{5})^{4}}. \]

Next, we simplify both the numerator and the denominator:

• The numerator becomes \[ (a^{3})^{4} = a^{12}. \]
• The denominator simplifies to \[ (-3)^{4} (b^{5})^{4} = 81 b^{20}. \]

Combining the simplified numerator and denominator, we have: \[ \frac{a^{12}}{81 b^{20}}. \]

Final Answer