Let's simplify the given expression and compare it with the provided options step by step.
The expression to simplify is:
[tex]\[ \sqrt{\frac{16n}{m^3}}. \][/tex]
### Step 1: Simplify the Expression
To simplify the expression [tex]\(\sqrt{\frac{16n}{m^3}}\)[/tex], we start by simplifying under the square root:
[tex]\[ \sqrt{\frac{16n}{m^3}} = \frac{\sqrt{16n}}{\sqrt{m^3}}. \][/tex]
Next, we simplify each part separately. First, the numerator [tex]\(\sqrt{16n}\)[/tex]:
[tex]\[ \sqrt{16n} = \sqrt{16} \cdot \sqrt{n} = 4 \cdot \sqrt{n}. \][/tex]
Second, the denominator [tex]\(\sqrt{m^3}\)[/tex]:
[tex]\[ \sqrt{m^3} = \sqrt{m^2 \cdot m} = \sqrt{m^2} \cdot \sqrt{m} = m \cdot \sqrt{m} = m^{3/2}. \][/tex]
Now, we combine both simplified parts:
[tex]\[ \sqrt{\frac{16n}{m^3}} = \frac{4 \cdot \sqrt{n}}{m^{3/2}} = \frac{4\sqrt{n}}{m^{3/2}}. \][/tex]
### Step 2: Compare with Options
Now, let's compare this simplified form with the given options one by one.
Option 1: [tex]\(\frac{4 \sqrt{m n}}{m^2}\)[/tex]
Rewrite the expression:
[tex]\[ \frac{4 \sqrt{m n}}{m^2} = \frac{4 \sqrt{m} \cdot \sqrt{n}}{m^2} = 4 \cdot \frac{\sqrt{m}}{m^2} \cdot \sqrt{n} = 4 \cdot \frac{1}{m^{3/2}} \cdot \sqrt{n} = \frac{4 \sqrt{n}}{m^{3/2}}. \][/tex]
This matches our simplified form.
Option 2: [tex]\(\frac{\sqrt{m n}}{4 m}\)[/tex]
Rewrite the expression:
[tex]\[ \frac{\sqrt{m n}}{4 m} = \frac{\sqrt{m} \cdot \sqrt{n}}{4 m} = \frac{\sqrt{n}}{4} \cdot \frac{\sqrt{m}}{m} = \frac{\sqrt{n}}{4} \cdot \frac{1}{\sqrt{m}} = \frac{\sqrt{n}}{4 \sqrt{m}}. \][/tex]
This does not match our simplified form.
Option 3: [tex]\(\frac{4 \sqrt{m n}}{n^2}\)[/tex]
Rewrite the expression:
[tex]\[ \frac{4 \sqrt{m n}}{n^2} = \frac{4 \sqrt{m} \cdot \sqrt{n}}{n^2} = 4 \cdot \frac{\sqrt{m}}{n^2} \cdot \sqrt{n} = 4 \cdot \frac{\sqrt{m}}{n^{3/2}}. \][/tex]
This does not match our simplified form.
Option 4: [tex]\(\frac{4 \sqrt{m n}}{m}\)[/tex]
Rewrite the expression:
[tex]\[ \frac{4 \sqrt{m n}}{m} = \frac{4 \sqrt{m} \cdot \sqrt{n}}{m} = 4 \cdot \sqrt{n} \cdot \frac{\sqrt{m}}{m} = 4 \cdot \sqrt{n} \cdot \frac{1}{\sqrt{m}} = 4 \cdot \frac{\sqrt{n}}{\sqrt{m}} = \frac{4 \sqrt{n}}{\sqrt{m}}. \][/tex]
This does not match our simplified form.
### Conclusion
Out of all the given options, the only one that matches our simplified form of [tex]\(\sqrt{\frac{16n}{m^3}} = \frac{4 \sqrt{n}}{m^{3/2}}\)[/tex] is:
[tex]\(\boxed{\frac{4 \sqrt{m n}}{m^2}}\)[/tex].
