To solve for the quotient of [tex]\( \frac{20 \sqrt{z^6}}{\sqrt{16 z^7}} \)[/tex] in simplest radical form, follow these steps:
1. Simplify the numerator [tex]\( 20 \sqrt{z^6} \)[/tex]:
- First, simplify [tex]\( \sqrt{z^6} \)[/tex]. Since [tex]\( z^6 \)[/tex] is a perfect square, [tex]\( \sqrt{z^6} = z^3 \)[/tex].
- Thus, the numerator becomes [tex]\( 20 z^3 \)[/tex].
2. Simplify the denominator [tex]\( \sqrt{16 z^7} \)[/tex]:
- Separate the product inside the square root: [tex]\( \sqrt{16 z^7} = \sqrt{16} \cdot \sqrt{z^7} \)[/tex].
- [tex]\( \sqrt{16} = 4 \)[/tex] because [tex]\( 16 \)[/tex] is a perfect square.
- To simplify [tex]\( \sqrt{z^7} \)[/tex]: rewrite [tex]\( z^7 \)[/tex] as [tex]\( z^6 \cdot z \)[/tex]. Thus, [tex]\( \sqrt{z^7} = \sqrt{z^6} \cdot \sqrt{z} = z^3 \cdot \sqrt{z} \)[/tex].
- Therefore, the denominator becomes [tex]\( 4 z^3 \sqrt{z} \)[/tex].
3. Form the quotient:
- Now, we divide the simplified numerator by the simplified denominator:
[tex]\[
\frac{20 z^3}{4 z^3 \sqrt{z}}
\][/tex]
4. Simplify the quotient:
- Simplify the constants: [tex]\( \frac{20}{4} = 5 \)[/tex].
- Simplify the powers: [tex]\( \frac{z^3}{z^3 \sqrt{z}} = \frac{z^3}{z^3 \cdot z^{1/2}} = \frac{1}{z^{1/2}} \)[/tex] since [tex]\( z^3 \)[/tex] cancels out.
5. Express the simplified form:
- Recognize that [tex]\( \frac{1}{z^{1/2}} = \frac{1}{\sqrt{z}} \)[/tex].
Thus, the quotient of [tex]\( \frac{20 \sqrt{z^6}}{\sqrt{16 z^7}} \)[/tex] simplifies to [tex]\( \frac{5}{\sqrt{z}} \)[/tex].
The correct answer is:
[tex]\[ \frac{5}{\sqrt{z}} \][/tex]
