To determine the quotient of \( \frac{20 \sqrt{z^6}}{\sqrt{16 z^7}} \) in simplest radical form, follow these steps:
1. Simplify the numerator:
[tex]\[ 20 \sqrt{z^6} \][/tex]
Since \( \sqrt{z^6} = z^3 \), the numerator becomes:
[tex]\[ 20 z^3 \][/tex]
2. Simplify the denominator:
[tex]\[ \sqrt{16 z^7} \][/tex]
Recognize that \( 16 = 4^2 \) and \( z^7 = z^6 \times z \). Therefore, we can rewrite the expression under the square root as:
[tex]\[ \sqrt{16 z^7} = \sqrt{(4^2) (z^6 \times z)} = \sqrt{(4^2 z^6) \cdot z} = 4 \sqrt{z^6 \cdot z} \][/tex]
Given \( \sqrt{z^6 \cdot z} = \sqrt{z^7} = z^{7/2} \), the denominator simplifies to:
[tex]\[ 4 z^{7/2} \][/tex]
3. Form the quotient:
[tex]\[ \frac{20 z^3}{4 z^{7/2}} \][/tex]
4. Simplify the coefficient:
[tex]\[ \frac{20}{4} = 5 \][/tex]
Hence, the expression becomes:
[tex]\[ \frac{5 z^3}{z^{7/2}} \][/tex]
5. Simplify the expression with exponents:
Recall that \(z^3 = z^{6/2}\). Rewrite the expression as:
[tex]\[ \frac{5 z^{6/2}}{z^{7/2}} \][/tex]
Subtract the exponents:
[tex]\[ 5 z^{(6/2 - 7/2)} = 5 z^{-1/2} = 5 \cdot \frac{1}{z^{1/2}} \][/tex]
6. Rationalize the expression:
Recall that \( \frac{1}{z^{1/2}} = \frac{1}{\sqrt{z}} \), so we get:
[tex]\[ 5 \cdot \frac{1}{\sqrt{z}} = \frac{5}{\sqrt{z}} \][/tex]
Therefore, the quotient of \( 20 \sqrt{z^6} \div \sqrt{16 z^7} \) in simplest radical form is:
[tex]\[ \boxed{\frac{5}{\sqrt{z}}} \][/tex]
