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Sagot :
Of course! Let's reduce the given square roots into their respective mixed radicals step by step.
### (i) Reducing [tex]\(\sqrt{45}\)[/tex]
1. Identify the factors of the number 45 that include a perfect square.
The number 45 can be decomposed into [tex]\(45 = 9 \times 5\)[/tex]. Here, 9 is a perfect square.
2. Express [tex]\(\sqrt{45}\)[/tex] in terms of the perfect square factor.
We can write:
[tex]\[ \sqrt{45} = \sqrt{9 \times 5} \][/tex]
3. Separate the square root of the perfect square and the remaining factor.
Using the property [tex]\(\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}\)[/tex], we get:
[tex]\[ \sqrt{45} = \sqrt{9} \times \sqrt{5} \][/tex]
4. Simplify the square root of the perfect square.
Since [tex]\(\sqrt{9} = 3\)[/tex], we have:
[tex]\[ \sqrt{45} = 3 \sqrt{5} \][/tex]
So, the mixed radical form of [tex]\(\sqrt{45}\)[/tex] is [tex]\(3\sqrt{5}\)[/tex].
### (ii) Reducing [tex]\(\sqrt{300}\)[/tex]
1. Identify the factors of the number 300 that include a perfect square.
The number 300 can be decomposed into [tex]\(300 = 100 \times 3\)[/tex]. Here, 100 is a perfect square.
2. Express [tex]\(\sqrt{300}\)[/tex] in terms of the perfect square factor.
We can write:
[tex]\[ \sqrt{300} = \sqrt{100 \times 3} \][/tex]
3. Separate the square root of the perfect square and the remaining factor.
Using the property [tex]\(\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}\)[/tex], we get:
[tex]\[ \sqrt{300} = \sqrt{100} \times \sqrt{3} \][/tex]
4. Simplify the square root of the perfect square.
Since [tex]\(\sqrt{100} = 10\)[/tex], we have:
[tex]\[ \sqrt{300} = 10 \sqrt{3} \][/tex]
So, the mixed radical form of [tex]\(\sqrt{300}\)[/tex] is [tex]\(10\sqrt{3}\)[/tex].
### Final Answer
- The mixed radical form of [tex]\(\sqrt{45}\)[/tex] is [tex]\(3\sqrt{5}\)[/tex].
- The mixed radical form of [tex]\(\sqrt{300}\)[/tex] is [tex]\(10\sqrt{3}\)[/tex].
### (i) Reducing [tex]\(\sqrt{45}\)[/tex]
1. Identify the factors of the number 45 that include a perfect square.
The number 45 can be decomposed into [tex]\(45 = 9 \times 5\)[/tex]. Here, 9 is a perfect square.
2. Express [tex]\(\sqrt{45}\)[/tex] in terms of the perfect square factor.
We can write:
[tex]\[ \sqrt{45} = \sqrt{9 \times 5} \][/tex]
3. Separate the square root of the perfect square and the remaining factor.
Using the property [tex]\(\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}\)[/tex], we get:
[tex]\[ \sqrt{45} = \sqrt{9} \times \sqrt{5} \][/tex]
4. Simplify the square root of the perfect square.
Since [tex]\(\sqrt{9} = 3\)[/tex], we have:
[tex]\[ \sqrt{45} = 3 \sqrt{5} \][/tex]
So, the mixed radical form of [tex]\(\sqrt{45}\)[/tex] is [tex]\(3\sqrt{5}\)[/tex].
### (ii) Reducing [tex]\(\sqrt{300}\)[/tex]
1. Identify the factors of the number 300 that include a perfect square.
The number 300 can be decomposed into [tex]\(300 = 100 \times 3\)[/tex]. Here, 100 is a perfect square.
2. Express [tex]\(\sqrt{300}\)[/tex] in terms of the perfect square factor.
We can write:
[tex]\[ \sqrt{300} = \sqrt{100 \times 3} \][/tex]
3. Separate the square root of the perfect square and the remaining factor.
Using the property [tex]\(\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}\)[/tex], we get:
[tex]\[ \sqrt{300} = \sqrt{100} \times \sqrt{3} \][/tex]
4. Simplify the square root of the perfect square.
Since [tex]\(\sqrt{100} = 10\)[/tex], we have:
[tex]\[ \sqrt{300} = 10 \sqrt{3} \][/tex]
So, the mixed radical form of [tex]\(\sqrt{300}\)[/tex] is [tex]\(10\sqrt{3}\)[/tex].
### Final Answer
- The mixed radical form of [tex]\(\sqrt{45}\)[/tex] is [tex]\(3\sqrt{5}\)[/tex].
- The mixed radical form of [tex]\(\sqrt{300}\)[/tex] is [tex]\(10\sqrt{3}\)[/tex].
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