We will learn how to find the integer and decimal parts of numbers containing radicals and use them to calculate the values of expressions.
Understanding Integer and Decimal Parts
If the integer part of a number a containing a radical is n, its decimal part is a − n.
Example: The integer part of √5 is 2 (because 2 < √5 < 3), and its decimal part is √5 − 2
📘 Example ①
Find the integer part a and decimal part b of √7.
Solution: Since 2² = 4 and 3² = 9, 2 < √7 < 3. The integer part is a = 2, and the decimal part is b = √7 − 2
Find the integer part a and decimal part b of √7.
Solution: Since 2² = 4 and 3² = 9, 2 < √7 < 3. The integer part is a = 2, and the decimal part is b = √7 − 2
📘 Example ②
In the example above, find the values of a + b and a×b.
Solution: a + b = 2 + (√7 − 2) = √7, a×b = 2(√7 − 2) = 2√7 − 4
In the example above, find the values of a + b and a×b.
Solution: a + b = 2 + (√7 − 2) = √7, a×b = 2(√7 − 2) = 2√7 − 4
💡 Key Points
- First check which consecutive integers contain the number
- Decimal part = original number − integer part
- Carefully calculate the value of expressions using the integer and decimal parts
Practice Problems
- Find the integer and decimal parts of √10.
- If the integer part of √3 is a and the decimal part is b, find the value of a + b².
Answers and Explanations
- Answer: Since 3 < √10 < 4, the integer part is 3 and the decimal part is √10 − 3
- Answer: Since 1 < √3 < 2, a = 1 and b = √3 − 1. a + b² = 1 + (√3 − 1)² = 1 + 3 − 2√3 + 1 = 5 − 2√3