\begin{array}{l}
\text{16. } 5(\sqrt{m} - n) = 5m - 5n \\
\text{17. } \sqrt{2} \times \sqrt{9} = \sqrt{9} \times \sqrt{2} \\
\text{18. } 9\sqrt{2} + \sqrt{3} \times 139 \times \sqrt{2} \\
\text{19. } \frac{\sqrt{a}}{2b} = \frac{\sqrt{9}}{\sqrt{\frac{2}{5}}} \\
\end{array}



Answer :

Let's analyze and solve each mathematical expression given.

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Question 16:

[tex]\[ 5(\sqrt{m} - n) = 5m - 5n \][/tex]

First, we need to understand the properties of square roots and basic algebra:

1. Distribute 5 within the parentheses on the left-hand side.

[tex]\[ 5 \cdot (\sqrt{m}) - 5 \cdot n = 5m - 5n \][/tex]

So, we get:

[tex]\[ 5\sqrt{m} - 5n = 5m - 5n \][/tex]

Now we compare both sides:
- Left side: [tex]\( 5\sqrt{m} - 5n \)[/tex]
- Right side: [tex]\( 5m - 5n \)[/tex]

For these expressions to be equal, [tex]\(5\sqrt{m} = 5m\)[/tex].

Therefore, this equation will be true only if:

[tex]\[ \sqrt{m} = m \][/tex]

An equation is needed to understand if there is a value satisfying the equality.

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Question 17:

[tex]\[ \sqrt{02} \times \sqrt{9} = \sqrt{9} \times \sqrt{52} \][/tex]

This needs simplification and correct interpretation of given mathematical notation:
- Note: Assume [tex]\(\sqrt{02}\)[/tex] is [tex]\(\sqrt{2} \text{ typo corrected}\)[/tex].
- The properties of square roots help to simplify this.
Applying property: [tex]\(\sqrt{a} \times \sqrt{b} = \sqrt{ab}\)[/tex], combine expression inside the square root.

1. For Left side:

[tex]\[ \sqrt{2} \times \sqrt{9} = \sqrt{2 \times 9} = \sqrt{18} \][/tex]

2. The right side [tex]\( \sqrt{9} \times \sqrt{52} = \sqrt{9 \times 52} = \sqrt{468}\)[/tex]


Which yields:

[tex]\[ \sqrt{18} \neq \sqrt{468} \][/tex]

Normally the two sides should simplify based on specific context, investigation needed.

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Question 18:

[tex]\[ 9\sqrt{2} + \sqrt{3} \times 139 \times \sqrt{2} \][/tex]

Break down and simplify:

1. Consider multiplication:

[tex]\[ 139 \sqrt{2} \][/tex]

2. Include terms:

[tex]\[ 9\sqrt{2} + (\sqrt{3} \times 139\sqrt{2}) \][/tex]

Without knowing [tex]\(\sqrt{}3\)[/tex] simplifying or combining:
Final: [tex]\( 9\sqrt{2} + 139 \sqrt{6} \)[/tex] (irregular combine due to significant sqrt() undefined)

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Question 19:

[tex]\[ \frac{\sqrt{a}}{2b} = \frac{\sqrt{9}}{\sqrt{2/5}} \][/tex]

Solve step-by-step:

1. Simplifying right fraction side separately:

[tex]\[ \sqrt{9} = 3, \quad \sqrt{\frac{2}{5}} \Rightarrow \sqrt{2} / \sqrt{5}\rightarrow multiply \sqrt{a} \][/tex]

Thus:

[tex]\[ \frac{3}{\left(\sqrt{2} / \sqrt{5}\right)} = \frac{3}{\left(\sqrt{2}\)8 / \sqrt{5} )} \][/tex]

Result expression preferred manageable re-examined form:

Summarized: original compositions clarified.