Question

please answer both the question​

Answer 1

Questions

1. Arrange ∛4, √3 and ∜6 in ascending order.

2. Are the square roots of all positive integers irrational? Justify your answer with examples.

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Answers

1. Let's take this step-wise to have a better understanding upon it.

Step 1: Find the LCM of the roots.

We have square root, cube root and fourth root given so we'll have to find the LCM of 2, 3 and 4.

$\begin{array}{c|c} \tt 2 & \sf{ 2,3,4} \\ \cline{1-2} \tt 2 & \sf { 1,3,2} \\ \cline{1-2} \tt 3 & \sf{ 1,3,1} \\ \cline{1-2} \tt & \sf{ 1,1,1} \\ \end{array}$

LCM of 2, 3 and 4 → 2×2×3 = 12

Step 2: Now with the obtained LCM convert the roots and make them equal. [Root shall be 12th root]

⇒ $\sqrt[3]{4} = \sqrt[3\times 4 ]{4^{3}}$

⇒ $\sqrt[3]{4} = \sqrt[12]{256}$

⇒ $\sqrt{3} = \sqrt[2\times 6 ]{3^{6}}$

⇒ $\sqrt{3} = \sqrt[12]{729}$

⇒ $\sqrt[4]{6} = \sqrt[4\times 3]{6^{3}}$

⇒ $\sqrt[4]{6} = \sqrt[12]{216}$

Now the roots of all the numbers are the same and now therefore we can arrange them in ascending order.

The ascending order of the Given is -: $\sqrt[12]{216} < \sqrt[12]{256} < \sqrt[12]{729}$

∴ The required answer is that the ascending order of the following numbers are -: ∜6 < ∛4 < √3

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2. No, square roots of all positive integers are not irrational numbers.

For example -:

(i) 4 is a positive integer and the square root of 4 is 2.

2 is a rational number and can be represented in p/q form where p and q are integers and q≠0.

(ii) Similarly let's take 256.

256 is a positive integer and the square root of 256 is 16.

16 is a rational number and can be represented in p/q form where p and q are integers and q≠0.

∴ Square roots of all positive integers are not always irrational numbers and can also appear to be rational..

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Answer 2

Answer 1) :-

→ ³√4, √3 , ⁴√6

→ (4)^1/3 , (3)^1/2 , (6)^1/4

taking LCM of 3, 2 and 4, which is 12 . Now Multiply and divide the powers of each terms by LCM,

→ (4)^(1/3 * 12/12) , (3)^(1/2 * 12/12) , (6)^(1/4 * 12/12)

→ (4)^(4/12) , (3)^(6/12) , (6)^(3/12)

→ (4⁴)^(1/12) , (3⁶)^(1/12) , (6³)^(1/12)

→ (256)^(1/12) , (729)^(1/12) , (216)^(1/12)

since power is same now, we can conclude that,

→ 216 < 256 < 729

or,

→ (216)^(1/12) < (256)^(1/12) < (729)^(1/12) .

or,

→ ⁴√6 < ³√4 < √3 . (Ans.)

Answer 2) :-

Square roots of all positive integers is not irrational .

for example :-

• √4 = 2
• √9 = 3
• √16 = 4
• √25 = 5 etc .

therefore, we can conclude that, square roots of all positive integers is not always irrational .

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