Questions: Use the quotient rule to simplify. Assume that all variables represent positive real numbers. -∛(z^8 / 64x^3) -∛(z^8 / 64x^3) = □ (Simplify your answer. Use integers or fractions for any numbers in the expression. Type an exact answer, using radicals as needed.

Solution Steps

To simplify the given expression using the quotient rule, we need to apply the cube root to both the numerator and the denominator separately. Then, we simplify the resulting expression.

We start with the expression \[ -\sqrt[3]{\frac{z^{8}}{64 x^{3}}}. \] Using the quotient rule for cube roots, we can separate the numerator and denominator: \[ -\sqrt[3]{\frac{z^{8}}{64 x^{3}}} = -\frac{\sqrt[3]{z^{8}}}{\sqrt[3]{64 x^{3}}}. \]

Next, we simplify the numerator and denominator:

• The numerator simplifies to \[ \sqrt[3]{z^{8}} = z^{\frac{8}{3}}. \]
• The denominator simplifies to \[ \sqrt[3]{64 x^{3}} = \sqrt[3]{64} \cdot \sqrt[3]{x^{3}} = 4x. \]

Now we can combine the simplified numerator and denominator: \[ -\frac{z^{\frac{8}{3}}}{4x}. \]

Final Answer