Solved Exercises on Operations with Scientific Notation in Grade 7

Master operations with scientific notation: multiplication, division, addition, subtraction, and mixed operations through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Multiplication in Scientific Notation
Exercise 1
Multiply: (3.2 × 10⁵) × (4.5 × 10³)
Definition:

Multiplication in scientific notation: Multiply coefficients and add exponents: \((a \times 10^m) \times (b \times 10^n) = (a \times b) \times 10^{m+n}\)

Multiplication method:
  1. Multiply the coefficients together
  2. Add the exponents
  3. Adjust the result to proper scientific notation if needed
Original Expression
(3.2 × 10⁵) × (4.5 × 10³)
Multiply Coefficients
3.2 × 4.5 = 14.4
Add Exponents
10⁵⁺³ = 10⁸
Step 1: Multiply coefficients

3.2 × 4.5 = 14.4

Step 2: Add exponents

10⁵ × 10³ = 10⁵⁺³ = 10⁸

Step 3: Combine results

14.4 × 10⁸

Step 4: Adjust to proper scientific notation

14.4 = 1.44 × 10¹, so 14.4 × 10⁸ = 1.44 × 10¹ × 10⁸ = 1.44 × 10⁹

(3.2 × 10⁵) × (4.5 × 10³) = 1.44 × 10⁹
Final answer:

(3.2 × 10⁵) × (4.5 × 10³) = 1.44 × 10⁹

Applied rules:

• Multiplication law: \(10^m \times 10^n = 10^{m+n}\)

• Coefficient adjustment: Ensure final coefficient is between 1 and 10

• Decimal movement: When adjusting coefficient, move decimal and adjust exponent accordingly

2 Division in Scientific Notation
Exercise 2
Divide: \(\frac{8.4 \times 10^7}{2.1 \times 10^4}\)
Definition:

Division in scientific notation: Divide coefficients and subtract exponents: \(\frac{a \times 10^m}{b \times 10^n} = \frac{a}{b} \times 10^{m-n}\)

Original Expression
\(\frac{8.4 \times 10^7}{2.1 \times 10^4}\)
Divide Coefficients
8.4 ÷ 2.1 = 4.0
Subtract Exponents
10⁷⁻⁴ = 10³
Step 1: Divide coefficients

\(\frac{8.4}{2.1} = 4.0\)

Step 2: Subtract exponents

\(\frac{10^7}{10^4} = 10^{7-4} = 10^3\)

Step 3: Combine results

4.0 × 10³

Step 4: Verify coefficient range

4.0 is between 1 and 10, so no adjustment needed

\(\frac{8.4 \times 10^7}{2.1 \times 10^4} = 4.0 \times 10^3\)
Final answer:

\(\frac{8.4 \times 10^7}{2.1 \times 10^4} = 4.0 \times 10^3\)

Applied rules:

• Division law: \(\frac{10^m}{10^n} = 10^{m-n}\)

• Coefficient division: Divide the numerical parts separately

• Verification: Ensure final coefficient is between 1 and 10

3 Addition with Same Exponents
Exercise 3
Add: (5.2 × 10⁶) + (3.8 × 10⁶)
Definition:

Addition in scientific notation: When exponents are the same, add coefficients: \((a \times 10^n) + (b \times 10^n) = (a + b) \times 10^n\)

Original Expression
(5.2 × 10⁶) + (3.8 × 10⁶)
Add Coefficients
5.2 + 3.8 = 9.0
Keep Exponent
× 10⁶
Step 1: Verify same exponents

Both terms have exponent 6, so we can add coefficients directly

Step 2: Add coefficients

5.2 + 3.8 = 9.0

Step 3: Keep the same exponent

9.0 × 10⁶

Step 4: Verify coefficient range

9.0 is between 1 and 10, so no adjustment needed

(5.2 × 10⁶) + (3.8 × 10⁶) = 9.0 × 10⁶
Final answer:

(5.2 × 10⁶) + (3.8 × 10⁶) = 9.0 × 10⁶

Applied rules:

• Same exponent rule: Add coefficients when exponents match

• Different exponents: Must convert to same exponent first

• Verification: Check that final coefficient is between 1 and 10

Scientific Notation Operations Summary
\((a \times 10^n) \times (b \times 10^m) = (a \times b) \times 10^{n+m}\)
Multiplication
Multiplication
\((a \times 10^m) \times (b \times 10^n) = (a \times b) \times 10^{m+n}\)
Multiply coefficients, add exponents
Division
\(\frac{a \times 10^m}{b \times 10^n} = \frac{a}{b} \times 10^{m-n}\)
Divide coefficients, subtract exponents
Addition/Subtraction
Convert to same exponent, then add/subtract coefficients
Must have same power of 10
Key definitions:

Scientific notation: Expresses numbers as \(a \times 10^n\) where \(1 \leq a < 10\)

Coefficient: The number 'a' in scientific notation (between 1 and 10)

Base: Always 10 in scientific notation

Exponent: The power 'n' indicates magnitude of the number

Operations: Special rules apply when performing arithmetic with scientific notation

Operation methodology:
  1. Multiplication: Multiply coefficients, add exponents
  2. Division: Divide coefficients, subtract exponents
  3. Addition/Subtraction: Convert to same exponent, then add/subtract coefficients
  4. Adjustment: Ensure final coefficient is between 1 and 10
Tip 1: For multiplication, add exponents; for division, subtract exponents.
Tip 2: Always check that your final coefficient is between 1 and 10.
Tip 3: When adding/subtracting, make sure both numbers have the same exponent first.
Tip 4: When adjusting coefficients, move the decimal and change the exponent oppositely.
Common errors: Forgetting to adjust coefficient after operations, incorrect exponent calculations, mixing up addition/subtraction with multiplication/division rules.
Exam preparation: Practice all four operations, master coefficient adjustments, understand when to convert exponents for addition/subtraction.
Essential formulas:

• Multiplication: \((a \times 10^m) \times (b \times 10^n) = (a \times b) \times 10^{m+n}\)

• Division: \(\frac{a \times 10^m}{b \times 10^n} = \frac{a}{b} \times 10^{m-n}\)

• Addition: \((a \times 10^n) + (b \times 10^n) = (a + b) \times 10^n\)

• Subtraction: \((a \times 10^n) - (b \times 10^n) = (a - b) \times 10^n\)

Solution: Exercises 4 to 5
4 Subtraction with Different Exponents
Exercise 4
Subtract: (7.5 × 10⁶) - (2.3 × 10⁵)
Definition:

Subtraction with different exponents: Convert both numbers to the same exponent (usually the larger one), then subtract coefficients.

Original Expression
(7.5 × 10⁶) - (2.3 × 10⁵)
Convert to Same Exponent
(7.5 × 10⁶) - (0.23 × 10⁶)
Subtract Coefficients
7.5 - 0.23 = 7.27
Step 1: Identify different exponents

6 ≠ 5, so we need to convert to the same exponent

Step 2: Convert to larger exponent

Convert 2.3 × 10⁵ to have exponent 6: 2.3 × 10⁵ = 0.23 × 10⁶

Step 3: Subtract coefficients

7.5 - 0.23 = 7.27

Step 4: Combine with exponent

7.27 × 10⁶

(7.5 × 10⁶) - (2.3 × 10⁵) = 7.27 × 10⁶
Final answer:

(7.5 × 10⁶) - (2.3 × 10⁵) = 7.27 × 10⁶

Applied rules:

• Same exponent requirement: Addition/subtraction requires same exponent

• Conversion method: Convert smaller exponent to match larger one

• Decimal movement: Moving decimal left increases exponent, moving right decreases it

5 Mixed Operations Problem
Exercise 5
Calculate: \(\frac{(6.0 \times 10^8) \times (2.5 \times 10^3)}{3.0 \times 10^6}\)
Definition:

Mixed operations: Perform operations in order following PEMDAS/BODMAS rules, applying scientific notation rules at each step.

Numerator
(6.0 × 10⁸) × (2.5 × 10³)
Calculate Numerator
15.0 × 10¹¹
Complete Division
5.0 × 10⁵
Step 1: Perform multiplication in numerator

(6.0 × 10⁸) × (2.5 × 10³) = (6.0 × 2.5) × 10⁸⁺³ = 15.0 × 10¹¹

Step 2: Adjust numerator coefficient

15.0 = 1.5 × 10¹, so 15.0 × 10¹¹ = 1.5 × 10¹²

Step 3: Perform division

\(\frac{1.5 \times 10^{12}}{3.0 \times 10^6} = \frac{1.5}{3.0} \times 10^{12-6} = 0.5 \times 10^6\)

Step 4: Adjust final coefficient

0.5 = 5.0 × 10⁻¹, so 0.5 × 10⁶ = 5.0 × 10⁵

\(\frac{(6.0 \times 10^8) \times (2.5 \times 10^3)}{3.0 \times 10^6} = 5.0 \times 10^5\)
Final answer:

\(\frac{(6.0 \times 10^8) \times (2.5 \times 10^3)}{3.0 \times 10^6} = 5.0 \times 10^5\)

Applied rules:

• Order of operations: Follow PEMDAS/BODMAS rules

• Sequential operations: Apply rules step by step

• Multiple adjustments: May need to adjust multiple times in complex problems

Detailed Scientific Notation Operations Guide
\((a \times 10^n) \times (b \times 10^m) = (a \times b) \times 10^{n+m}\)
Multiplication Rule
Key definitions:

Scientific notation: A way to express very large or very small numbers in the form \(a \times 10^n\) where \(1 \leq a < 10\) and \(n\) is an integer.

Coefficient: The number \(a\) in scientific notation (the part between 1 and 10).

Exponent: The integer \(n\) that tells us how many places the decimal was moved.

Operations: Special rules for performing arithmetic with numbers in scientific notation.

Complete operation methodology:
  1. Multiplication: Multiply coefficients and add exponents
  2. Division: Divide coefficients and subtract exponents
  3. Addition: Ensure same exponent, then add coefficients
  4. Subtraction: Ensure same exponent, then subtract coefficients
  5. Adjustment: Always ensure final coefficient is between 1 and 10
Tip 1: For multiplication, add exponents; for division, subtract the bottom exponent from the top exponent.
Tip 2: When adding or subtracting, always convert to the same exponent first.
Tip 3: If your coefficient is not between 1 and 10 after an operation, adjust it by moving the decimal.
Tip 4: Moving decimal left increases exponent; moving right decreases exponent.
Common errors: Forgetting to adjust coefficients after operations, incorrect exponent signs, not converting to same exponents for addition/subtraction, miscalculating when coefficients exceed 10.
Applications: Scientific calculations, engineering, astronomy, chemistry, physics, computer science, and any field dealing with very large or very small numbers.
Essential laws and formulas:

• Multiplication: \((a \times 10^m) \times (b \times 10^n) = (a \times b) \times 10^{m+n}\)

• Division: \(\frac{a \times 10^m}{b \times 10^n} = \frac{a}{b} \times 10^{m-n}\)

• Addition: \((a \times 10^n) + (b \times 10^n) = (a + b) \times 10^n\)

• Subtraction: \((a \times 10^n) - (b \times 10^n) = (a - b) \times 10^n\)

• Exponent rules: \(10^m \times 10^n = 10^{m+n}\), \(\frac{10^m}{10^n} = 10^{m-n}\)

Scientific Notation Operations Guide

📊
Multiplication

Multiply coefficients

Add exponents

Example: (2 × 10³) × (3 × 10⁴) = 6 × 10⁷

Division

Divide coefficients

Subtract exponents

Example: (6 × 10⁸) ÷ (2 × 10³) = 3 × 10⁵

Addition/Subtraction

Same exponent required

Add/subtract coefficients

Example: (4 × 10⁵) + (3 × 10⁵) = 7 × 10⁵

Adjustment Rules

• Coefficient must be 1 ≤ a < 10

• Move decimal left → increase exponent

• Move decimal right → decrease exponent

Questions & Answers

Question: When I multiply (3.2 × 10⁵) × (4.0 × 10³) and get 12.8 × 10⁸, why do I need to change it to 1.28 × 10⁹?

Answer: This is about maintaining proper scientific notation format! The definition requires that the coefficient be between 1 and 10 (1 ≤ a < 10).

When you get 12.8 × 10⁸:

  1. 12.8 is not between 1 and 10 (it's greater than 10)
  2. Rewrite 12.8 as 1.28 × 10¹
  3. Combine: (1.28 × 10¹) × 10⁸ = 1.28 × 10⁹

Always check that your final coefficient is in the proper range. This ensures consistency and makes comparisons easier.

Think of it as "standardizing" the format so everyone writes numbers the same way!

Question: When adding (5.2 × 10⁶) + (3.8 × 10⁵), why can't I just add the coefficients and exponents separately?

Answer: Addition and subtraction require the same exponent because you're combining like terms!

Think of it this way:

  • 5.2 × 10⁶ = 5,200,000
  • 3.8 × 10⁵ = 380,000

You can't add 5.2 + 3.8 directly because they represent different orders of magnitude.

Instead, convert to the same exponent:

  1. 3.8 × 10⁵ = 0.38 × 10⁶
  2. Now add: (5.2 × 10⁶) + (0.38 × 10⁶) = 5.58 × 10⁶

This is similar to how you need common denominators to add fractions!

Question: In division, why do we subtract the bottom exponent from the top exponent? For example, \(\frac{10^7}{10^3} = 10^4\)?

Answer: This follows from the fundamental property of exponents! Let's think about what division means:

\(\frac{10^7}{10^3}\) means \(\frac{10 \times 10 \times 10 \times 10 \times 10 \times 10 \times 10}{10 \times 10 \times 10}\)

When you cancel out three 10's from the numerator and denominator:

\(\frac{10 \times 10 \times 10 \times 10 \times 10 \times 10 \times 10}{10 \times 10 \times 10} = 10 \times 10 \times 10 \times 10 = 10^4\)

This gives us the general rule: \(\frac{10^m}{10^n} = 10^{m-n}\)

It's like having 7 tens in the numerator and removing 3 tens (denominator), leaving 4 tens!