Key Concepts from the Lecture

Example 1: Finding Consecutive Integers for the Square Root of 7

Objective: Determine between which two consecutive integers the square root of 7 lies.
Consecutive Integers: Two whole numbers that surround a given value.
Approach:
- Recognize that the square root of a number will be between the square roots of two perfect squares.
- Recall some perfect squares:
  - (2^2 = 4)
  - (3^2 = 9)
- Since 7 is between 4 and 9, the square root of 7 is between the square roots of these values.
- Conclusion:
  - (\sqrt{4} = 2)
  - (\sqrt{9} = 3)
- Therefore, the square root of 7 is between 2 and 3.

Objective: Determine between which two consecutive integers the cube root of 100 lies.
Approach:
- Recognize that the cube root of a number will be between the cube roots of two perfect cubes.
- Recall some perfect cubes:
  - (2^3 = 8)
  - (3^3 = 27)
  - (4^3 = 64)
  - (5^3 = 125)
- Since 100 is between 64 and 125, the cube root of 100 is between the cube roots of these values.
- Conclusion:
  - (\sqrt[3]{64} = 4)
  - (\sqrt[3]{125} = 5)
- Therefore, the cube root of 100 is between 4 and 5.

To find between which two integers a root lies, identify the perfect squares (or cubes) surrounding the given number.
Utilize knowledge of basic perfect squares and cubes to deduce the integers between which the root of a number lies.