1. If two irrational numbers radicals are in the form
(a + √b) and (c + √d),
estimate the value of each irrational number and compare them.
2. If two irrational numbers are in the form
√a and √b,
square each irrational number to get rid of the square root. Then, compare them.
Example 1 :
Compare (√3 + 5) and (3 + √5) and write <, >, or = in between them.
Answer :
Step 1 :
Approximate √3.
√3 is between 1 and 2
Step 2 :
Approximate √5.
√5 is between 2 and 3
Step 3 :
Use your approximations in the above steps to estimate the values of the given irrational numbers.
√3 + 5 is between 6 and 7
3 + √5 is between 5 and 6
Therefore,
√3 + 5 > 3 + √5
Example 2 :
Compare (√2 + 4) and (√2 + √25) and write <, >, or = in between them.
Answer :
Step 1 :
Approximate √2.
√2 is between 1 and 2
Step 2 :
Approximate √4.
√4 is equal 2
Step 3 :
Approximate √25.
√25 is equal 5
Step 4 :
Use your approximations in the above steps to estimate the values of the given irrational numbers.
√2 + 4 is between 5 and 6
√2 + √25 = √2 + 5
√2 + 5 is between 6 and 7
√2 + 4 < √2 + 5
Therefore,
√2 + 4 < √2 + √25
Example 3 :
Compare 4√2 and 3√3 and write <, >, or = in between them.
Answer :
Step 1 :
Square 4√2.
(4√2)^{2} = (4)^{2}(√2)^{2}
(4√2)^{2} = (16)(2)
(4√2)^{2} = 32 ----> (1)
Step 2 :
Square 3√3.
(3√3)^{2} = (3)^{2}(√3)^{2}
(3√3)^{2} = (9)(3)
(3√3)^{2} = 27 ----> (2)
Step 3 :
Comparing (1) and (2),
32 > 27 ----> 4√2 > 3√3
Example 4 :
Compare 3√8 and 4√5 and write < or > between them.
Answer :
Step 1 :
Take square to the number 3√8.
(3√8)^{2} = 3^{2}(√8)^{2}
(3√8)^{2} = 9(8)
(3√8)^{2} = 72 ----> (2)
Step 2 :
Take square to the number 4√5.
(4√5)^{2} = 4^{2}(√5)^{2}
(4√5)^{2} = 16(5)
(4√5)^{2} = 80 ----> (1)
Step 3 :
From (1) and (2), we get
72 < 80 ----> 3√8 < 4√5
Example 5 :
Compare (√12 + 6) and (12 + √6) and write <, >, or = in between them.
Answer :
Step 1 :
Approximate √12.
√12 is between 3 and 4
Step 2 :
Approximate √6.
√6 is between 2 and 3
Step 3 :
Use your approximations in the above steps to estimate the values of the given irrational numbers.
√12 + 6 is between 9 and 10
12 + √6 is between 12 and 14
Therefore,
√12 + 6 < 12 + √6
Example 6 :
Compare (√5 + 6) and (5 + √6) and write <, >, or = in between them.
Answer :
Step 1 :
Approximate √5.
√5 is between 2 and 3
Step 2 :
Approximate √6.
√6 is between 2 and 3
Step 3 :
Use your approximations in the above steps to estimate the values of the given irrational numbers.
√5 + 6 is between 8 and 9
5 + √6 is between 7 and 8
Therefore,
√5 + 6 > 5 + √6
Example 7 :
Compare (√3 + 3) and (√3 + √9) and write <, >, or = in between them.
Answer :
√3 + 3 ----(1)
√3 + √9 = √3 + 3 ----(2)
Comparing (1) and (2),
√3 + 3 = √3 + √9
Example 8 :
Compare (√3 + 3) and (√9 + √3) and write <, >, or = in between them.
Answer :
Step 1 :
First approximate √3.
√3 is between 1 and 2
Step 2 :
Then use your approximations to simplify the expressions.
√3 + 3 is between 4 and 5 ----(1)
Step 3 :
Find the value of √9.
√9 is equal to 3
Step 4 :
√9 + √3 = 3 + √3
3 + √3 is between 4 and 5 ----(2)
Step 5 :
From (1) and (2), we get,
√3 + 3 = 3 + √3
Therefore,
√3 + 3 = √9 + √3
Example 9 :
Compare √3 and 𝝅 and write <, >, or = in between them.
Answer :
Step 1 :
First approximate √3.
√3 is between 1 and 2
Step 2 :
Using calculator, approximate 𝝅.
𝝅 = 3.1415......
The value of 𝝅 is between 3 and 4.
Therefore,
√3 < 𝝅
Example 10 :
Compare (√2 + 5) and (𝝅 + 2) and write <, >, or = in between them.
Answer :
Step 1 :
First approximate √2.
√2 is between 1 and 2
Step 2 :
Then use your approximation to simplify the expression.
√2 + 5 is between 6 and 7 ----(1)
Step 3 :
The value of 𝝅 is between 3 and 4.
Step 4 :
Then use your approximation to simplify the expression.
𝝅 + 2 is between 5 and 6 ----(2)
Therefore,
√2 + 5 < 𝝅 + 2
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