Solving 6th-grade math problems on pages 30 and 31: Exponents and Common Bases

=> Check out the latest 6th-grade math solutions here: 6th-grade math solutions
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- Solving 6th-grade math problems on page 26 of Book Connect Knowledge - Exercise 7: Order of operations

Guide to solving sixth-grade math problems on pages 30 and 31 (Concise)

A. Synthesis of Knowledge

Overview

Note:
- When dividing twin exponents with the same base (excluding 0), keep the base unchanged and subtract the exponents.
- Every natural number can be written as the sum of exponents of 10.
Example:
- 5⁸ : 5³ = 5^8-3 = 5⁵
- 3255 = 3.1000 + 2.100 + 5.10 + 5.1 = 3.10³ + 2.10² + 5.10¹ + 5.10^0

B. Exercises and Guidance

1. Solve math problem 6 page 30, 31 exercise 67

Write the result of each calculation below in the form of an exponent:
a) 3⁸ : 3⁴
b) 10⁸ : 10²
c) a⁶ : a (a not equal to 0)
Solution:
a) 3⁸ : 3⁴ = 3^8-4 = 3
c) a⁶ : a = a^6-1 = a⁵ (a not equal to 0)

2. Solve math problem 6 page 30, 31 exercise 68

Calculate in two ways:
Method 1: Calculate the dividend, calculate the divisor, then calculate the quotient.
Method 2: Divide twin exponents with the same base and then calculate the result.
a) 2¹⁰ : 2⁸
b) 4⁶ : 4³
c) 8⁵ : 8⁴
d) 7⁴ : 7⁴
Solution:
Method 1
a) 1024 : 256 = 4
b) 4096 : 64 = 64
c) 32768 : 4096 = 8
d) 2401 : 2401 = 1
Method 2
a) 2¹⁰ : 2⁸ = 2^10–8 = 2² = 4
b) 4⁶ : 4³ = 4^6–3 = 4³ = 64
c) 8⁵ : 8⁴ = 8^5–4 = 8¹ = 8
d) 7⁴ : 7⁴ = 7^4–4 = 7⁰ = 1

3. Solve math problem 6 page 30, 31 exercise 69

Fill in the square with the letter C (correct) or the letter I (incorrect):

Solution:

4. Solve math problem 70 page 30 Math 6 Book 1

Write the numbers: 987; 2564; abcde in the form of the sum of exponents of 10
Solution:
987 = 9.10² + 8.10 + 7;
2564 = 2.10³ + 5.10² + 6.10 + 4;
abcde= a.10⁴ + b.10³ + c.10² + d.10 + e

5. Solve math problem 6 page 30, 31 exercise dividing twin exponents with a common base exercise 71

Find the natural number c, knowing that for every n in N* we have:
a) cⁿ = 1
b) cⁿ = 0.
Solution:
a) c = 1
b) c = 0

6. Solve math problem 6 page 30, 31 exercise 72

Perfect square numbers are numbers equal to the square of a natural number (e.g., 0, 1, 4, 9, 16…). Is each of the following sums a perfect square number?
a) 1³ + 2³
b) 1³ + 2³ + 3³
c) 1³ + 2³ + 3³ + 4³
Solution:
a) 1³ + 2³ = 1 + 8 = 9 = 3²
b) 1³ + 2³ + 3³ = 1 + 8 + 27 = 36 = 6²
c) 1³ + 2³ + 3³ + 4³ = 1 + 8 + 27 + 64 = 100 = 10²

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With a scientific and easily understandable presentation, solving math problems in Grade 6 Volume 1 page 30, 31 is a valuable reference material for students to study and review dividing two powers with the same base. Next, students can refer to the section Solving math problems in Grade 6 Volume 1 page 27, 28, 29 that has been solved earlier or preview the section Solving math problems in Grade 6 Volume 1 page 32, 33 to excel in the subject of Grade 6 Mathematics.