Key Concept :
To compare and order rational and irrational numbers, we can approximate irrational numbers as decimals.
Example 1 :
Compare (√3 + 5) and 22/3 and write < or > 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 + 5 is between 6 and 7
Step 2 :
Now, find the value of 22/3.
The approximate value of 22/3 is 7.33.
So, we have
(√3 + 5) < 22/3
Example 2 :
Compare (√2 + 4) and (2 + √4) and write < or > between them.
Answer :
Step 1 :
First approximate √2.
√2 is between 1 and 2
Next approximate √4.
√4 is equal 2
Step 2 :
Then use your approximations to simplify the expressions.
√2 + 4 is between 5 and 6.
2 + √4 is equal to 4
So, we have
√2 + 4 > 2 + √4
Example 3 :
Compare 4√5 and 3√9 and write < or > between them.
Answer :
Key Concept :
Square both the numbers and compare them.
Step 1 :
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 2 :
Take square to the number 3√3.
(3√3)^{2} = (3)^{2}(√9)^{2}
(3√3)^{2} = (9)(9)
(3√3)^{2} = 81 -----> (2)
Step 3 :
From (1) and (2), we get
80 < 81 -----> 4√5 < 3√9
Example 4 :
Compare (√3 + 3) and (√9 + √16) 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 6 and 7 -----(1)
Step 2 :
Find the value of √9.
√9 is equal to 3
Step 3 :
Find the value of √16.
√16 is equal to 4.
Step 4 :
√9 + √16 = 3 + 4
√9 + √16 = 7 -----(2)
Step 4 :
From (1) and (2), we get,
√3 + 3 < √9 + √16
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