Square and cube numbers - Multiplication by URBrainy.com - Free Printable

Problem Analysis:

The worksheet focuses on square numbers and their properties. Let's solve each part step by step.

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#### 1. Calculate the given expressions involving squares:

Example:
$$
3^2 + 4^2 = (3 \times 3) + (4 \times 4) = 9 + 16 = 25
$$

Instructions: The square of each number must be calculated first, then the results are added or subtracted as required.

##### Problem 1:
$$
3^2 + 5^2 = (3 \times 3) + (5 \times 5) = 9 + 25 = 34
$$

##### Problem 2:
$$
4^2 + 6^2 = (4 \times 4) + (6 \times 6) = 16 + 36 = 52
$$

##### Problem 3:
$$
7^2 + 7^2 = (7 \times 7) + (7 \times 7) = 49 + 49 = 98
$$

##### Problem 4:
$$
6^2 - 4^2 = (6 \times 6) - (4 \times 4) = 36 - 16 = 20
$$

##### Problem 5:
$$
10^2 - 7^2 = (10 \times 10) - (7 \times 7) = 100 - 49 = 51
$$

##### Problem 6:
$$
9^2 - 2^2 = (9 \times 9) - (2 \times 2) = 81 - 4 = 77
$$

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#### 2. Identify which numbers in the box are square numbers:

The numbers in the box are:
$$
169, 200, 1,000, 2,025, 18, 9
$$

A square number is a number that can be expressed as \( n^2 \) for some integer \( n \).

- 169:
$$
169 = 13^2 \quad (\text{since } 13 \times 13 = 169)
$$

- 200:
This is not a perfect square because there is no integer \( n \) such that \( n^2 = 200 \).

- 1,000:
This is not a perfect square because there is no integer \( n \) such that \( n^2 = 1,000 \).

- 2,025:
$$
2,025 = 45^2 \quad (\text{since } 45 \times 45 = 2,025)
$$

- 18:
This is not a perfect square because there is no integer \( n \) such that \( n^2 = 18 \).

- 9:
$$
9 = 3^2 \quad (\text{since } 3 \times 3 = 9)
$$

Thus, the square numbers in the box are:
$$
\boxed{169, 2,025, 9}
$$

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#### 3. Find the square root of 169:

The square root of a number \( x \) is a value \( y \) such that \( y^2 = x \). For 169:
$$
\sqrt{169} = 13 \quad (\text{since } 13 \times 13 = 169)
$$

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#### 4. Express the given totals as the sum of two square numbers:

We need to find pairs of square numbers that add up to the given totals.

##### Problem 9: 74
We need \( a^2 + b^2 = 74 \). Testing small integers:
- \( 7^2 = 49 \) and \( 5^2 = 25 \):
$$
49 + 25 = 74
$$
Thus, \( 74 = 7^2 + 5^2 \).

##### Problem 10: 85
We need \( a^2 + b^2 = 85 \). Testing small integers:
- \( 9^2 = 81 \) and \( 2^2 = 4 \):
$$
81 + 4 = 85
$$
Thus, \( 85 = 9^2 + 2^2 \).

##### Problem 11: 45
We need \( a^2 + b^2 = 45 \). Testing small integers:
- \( 6^2 = 36 \) and \( 3^2 = 9 \):
$$
36 + 9 = 45
$$
Thus, \( 45 = 6^2 + 3^2 \).

##### Problem 12: 181
We need \( a^2 + b^2 = 181 \). Testing small integers:
- \( 13^2 = 169 \) and \( 6^2 = 36 \):
$$
169 + 12 = 181
$$
Thus, \( 181 = 13^2 + 6^2 \).

##### Problem 13: 52
We need \( a^2 + b^2 = 52 \). Testing small integers:
- \( 6^2 = 36 \) and \( 4^2 = 16 \):
$$
36 + 16 = 52
$$
Thus, \( 52 = 6^2 + 4^2 \).

##### Problem 14: 113
We need \( a^2 + b^2 = 113 \). Testing small integers:
- \( 10^2 = 100 \) and \( 7^2 = 49 \):
$$
100 + 13 = 113
$$
Thus, \( 113 = 10^2 + 7^2 \).

##### Problem 15: 40
We need \( a^2 + b^2 = 40 \). Testing small integers:
- \( 6^2 = 36 \) and \( 2^2 = 4 \):
$$
36 + 4 = 40
$$
Thus, \( 40 = 6^2 + 2^2 \).

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Final Answers:

1. \( 3^2 + 5^2 = 34 \)
2. \( 4^2 + 6^2 = 52 \)
3. \( 7^2 + 7^2 = 98 \)
4. \( 6^2 - 4^2 = 20 \)
5. \( 10^2 - 7^2 = 51 \)
6. \( 9^2 - 2^2 = 77 \)
7. Square numbers in the box: \( \boxed{169, 2,025, 9} \)
8. \( \sqrt{169} = 13 \)
9. \( 74 = 7^2 + 5^2 \)
10. \( 85 = 9^2 + 2^2 \)
11. \( 45 = 6^2 + 3^2 \)
12. \( 181 = 13^2 + 6^2 \)
13. \( 52 = 6^2 + 4^2 \)
14. \( 113 = 10^2 + 7^2 \)
15. \( 40 = 6^2 + 2^2 \)