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.
---
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.
---
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)
---
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.
---
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.
---
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.
---
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)
---
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.
