Find how many non- squares natural number lie between the following pair. 7² and 8²

To find the number of non-square natural numbers between \( 7^2 \) and \( 8^2 \), let's proceed with the following steps:

1. **Calculate the squares**:
   \[
   7^2 = 49
   \]
   \[
   8^2 = 64
   \]

2. **List the natural numbers between 49 and 64**:
   The natural numbers between 49 and 64 are:
   \[
   50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63
   \]

   So, there are \( 64 - 49 - 1 = 14 \) natural numbers between them.

3. **Identify the perfect squares in this range**:
   The perfect squares in this range are \( 7^2 = 49 \) and \( 8^2 = 64 \), but they are not included because we're only interested in the numbers between them. For the numbers between 49 and 64, we only need to check for any perfect squares in this interval:

   - The next perfect square after 49 is \( 8^2 = 64 \).

   Since there are no other squares between \( 49 \) and \( 64 \), all 14 numbers are non-squares.

4. **Conclusion**:
   The number of non-square natural numbers between \( 7^2 \) and \( 8^2 \) is \( 14 \).

Would you like further details or have any questions?

Here are some related questions for further exploration:
1. What is the number of non-square natural numbers between \( 10^2 \) and \( 11^2 \)?
2. How many non-square natural numbers are there between \( 5^2 \) and \( 6^2 \)?
3. Can you find the number of perfect squares between \( 3^2 \) and \( 9^2 \)?
4. How many non-square natural numbers lie between \( 9^2 \) and \( 12^2 \)?
5. What is the pattern in the number of non-square natural numbers between \( n^2 \) and \( (n+1)^2 \)?

**Tip**: The difference between consecutive squares \( n^2 \) and \( (n+1)^2 \) is always \( 2n + 1 \).

Discover how many non-square natural numbers lie between 7² and 8² with a detailed step-by-step solution. Learn about perfect squares, natural numbers, and the formula for calculating the difference between consecutive squares. Suitable for high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation