https://file.aiquickdraw.com/mathbot/file/1730140070979dyd84nol.jpg

Let's solve each of the equations from the image:

### Problem 13
\[
1^2 + 2^2 + 3^2 + 4^2 = 1 + 4 + 9 + 16 = 30
\]
**Answer:** 30

### Problem 14
\[
3^3 \div 9 + 128 \div 4^2
\]
Calculate each part:
- \( 3^3 = 27 \) and \( 27 \div 9 = 3 \)
- \( 4^2 = 16 \) and \( 128 \div 16 = 8 \)
\[
3 + 8 = 11
\]
**Answer:** 11

### Problem 15
\[
(5 + 1)^2 + 107
\]
Calculate each part:
- \( 5 + 1 = 6 \) and \( 6^2 = 36 \)
\[
36 + 107 = 143
\]
**Answer:** 143

### Problem 16
\[
\frac{63}{3^2} + 5^3 - 4^2
\]
Calculate each part:
- \( 3^2 = 9 \) and \( 63 \div 9 = 7 \)
- \( 5^3 = 125 \)
- \( 4^2 = 16 \)
\[
7 + 125 - 16 = 116
\]
**Answer:** 116

### Problem 17
\[
2(40 + 21 \cdot 6) - 10^2
\]
Calculate each part:
- \( 21 \cdot 6 = 126 \)
- \( 40 + 126 = 166 \)
- \( 2 \cdot 166 = 332 \)
- \( 10^2 = 100 \)
\[
332 - 100 = 232
\]
**Answer:** 232

### Problem 18
\[
(9^2 + 8^2) - 144 \div 12
\]
Calculate each part:
- \( 9^2 = 81 \) and \( 8^2 = 64 \)
- \( 81 + 64 = 145 \)
- \( 144 \div 12 = 12 \)
\[
145 - 12 = 133
\]
**Answer:** 133

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Would you like further details on any of these solutions, or have additional questions?

Here are five related questions to extend your understanding:
1. What is the general formula for calculating the sum of squares?
2. How does exponentiation impact order of operations in a complex expression?
3. How can you simplify expressions that involve both division and exponents?
4. What steps can help avoid errors in multi-step arithmetic problems?
5. How does the distributive property apply in equations like Problem 17?

**Tip:** Always follow the order of operations (PEMDAS/BODMAS) to ensure accurate results.

This guide provides detailed solutions for a set of arithmetic problems involving exponents, division, and addition. The problems focus on using the order of operations to solve expressions with multiple terms, squares, and divisions. Suitable for Grades 5-8 students.

Math Problem Statement

Solution

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Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

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