Question

enter the correct answer in the box.
if ( z > 0 ), what is the quotient of ( 20sqrt{z^6} div sqrt{16z^7} ) in simplest radical form? if necessary, rationalize the denominator.

Explanation:

Step1: Rewrite division as fraction

Rewrite \(20\sqrt{z^{6}}\div\sqrt{16z^{7}}\) as \(\frac{20\sqrt{z^{6}}}{\sqrt{16z^{7}}}\).

Step2: Use quotient rule for radicals

The quotient rule for radicals states that \(\frac{\sqrt{a}}{\sqrt{b}}=\sqrt{\frac{a}{b}}\) (for \(a\geq0,b > 0\)). So we can rewrite the expression as \(20\sqrt{\frac{z^{6}}{16z^{7}}}\).

Step3: Simplify the fraction inside the radical

Simplify \(\frac{z^{6}}{16z^{7}}\) using the rule \(\frac{a^{m}}{a^{n}}=a^{m - n}\). Here, \(m = 6\), \(n=7\), so \(\frac{z^{6}}{z^{7}}=z^{6-7}=z^{-1}=\frac{1}{z}\). So the expression becomes \(20\sqrt{\frac{1}{16z}}\).

Step4: Simplify the square root

We know that \(\sqrt{\frac{1}{16z}}=\frac{\sqrt{1}}{\sqrt{16z}}=\frac{1}{4\sqrt{z}}\) (since \(\sqrt{16}=4\)). So now the expression is \(20\times\frac{1}{4\sqrt{z}}\).

Step5: Simplify the coefficient

Simplify \(20\times\frac{1}{4}=\frac{20}{4} = 5\). So we have \(\frac{5}{\sqrt{z}}\).

Step6: Rationalize the denominator

To rationalize the denominator, multiply the numerator and denominator by \(\sqrt{z}\). So \(\frac{5\times\sqrt{z}}{\sqrt{z}\times\sqrt{z}}=\frac{5\sqrt{z}}{z}\) (since \(\sqrt{z}\times\sqrt{z}=z\) for \(z>0\)).

Answer:

\(\frac{5\sqrt{z}}{z}\)