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
So,
√3 + 5 > 3 + √5
Example 2 :
Compare (√2 + 4) and (2 + √4) 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 :
Use your approximations in the above steps to estimate the values of the given irrational numbers.
√2 + 4 is between 5 and 6
2 + √4 is equal to 4
So,
√2 + 4 > 2 + √4
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 (√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
So,
√12 + 6 < 12 + √6
Example 5 :
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
So,
√5 + 6 > 5 + √6
Example 6 :
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
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