Solved Exercises on Operations with Real Numbers in Grade 8

Master operations with real numbers: addition, subtraction, multiplication, division, and order of operations through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Mixed Operations with Integers
Exercise 1
Calculate: (-15) + 8 - (-12) + (-7). Show all steps and explain the rules for adding and subtracting integers.
Definition:

Integer: A whole number that can be positive, negative, or zero (..., -3, -2, -1, 0, 1, 2, 3, ...).

Real Number: Any number that can be found on the number line, including integers, fractions, and decimals.

Sign Rules: Rules governing how positive and negative numbers interact in operations.

Integer operation methodology:
  1. Convert subtractions: Change subtraction to addition using opposite
  2. Group by sign: Organize positive and negative numbers separately
  3. Apply addition rules: Same signs add, different signs subtract
  4. Determine final sign: Based on the larger absolute value
  5. Verify result: Check with number line or estimation
Step 1: Convert subtractions to additions

Original expression: (-15) + 8 - (-12) + (-7)

Rule: Subtracting a negative is the same as adding a positive

So -(-12) = +12

New expression: (-15) + 8 + 12 + (-7)

Step 2: Group positive and negative numbers

Positive numbers: 8, 12

Negative numbers: -15, -7

Step 3: Add numbers with the same sign

Positive sum: 8 + 12 = 20

Negative sum: (-15) + (-7) = -22

Step 4: Add the results

20 + (-22) = 20 - 22 = -2

When adding numbers with different signs, subtract absolute values and take the sign of the larger absolute value

Step 5: Verify with number line approach

Start at -15, move +8 to -7, move +12 to +5, move -7 to -2

Final position: -2 ✓

Final answer:

(-15) + 8 - (-12) + (-7) = -2

Applied rules:

• Subtraction rule: a - b = a + (-b)

• Double negative: -(-a) = +a

• Same signs: Add absolute values, keep common sign

• Different signs: Subtract absolute values, keep sign of larger

Start: -15 Move +8 Move +12 Move -7 Final: -2 -20 -15 -10 -5 0 5 10 15 (-15) + 8 + 12 + (-7) = -2 Integer Addition and Subtraction: Number Line Method
2 Fraction and Decimal Operations
Exercise 2
Calculate: (3/4) × (-2/5) ÷ (6/7). Express your answer in simplest form.
Definition:

Rational Number: A number that can be expressed as a/b where a and b are integers and b ≠ 0.

Equivalent Fractions: Fractions that represent the same value (e.g., 1/2 = 2/4).

Reciprocal: For fraction a/b, the reciprocal is b/a (used in division).

Step 1: Convert division to multiplication

Dividing by a fraction is the same as multiplying by its reciprocal

(3/4) × (-2/5) ÷ (6/7) = (3/4) × (-2/5) × (7/6)

Step 2: Multiply the numerators and denominators

Multiply all numerators together: 3 × (-2) × 7 = -42

Multiply all denominators together: 4 × 5 × 6 = 120

Result: -42/120

Step 3: Simplify the fraction

Find the GCD of 42 and 120:

42 = 2 × 3 × 7

120 = 2³ × 3 × 5

GCD = 2 × 3 = 6

Step 4: Divide both numerator and denominator by GCD

-42 ÷ 6 = -7

120 ÷ 6 = 20

Simplified: -7/20

Step 5: Verify the simplification

GCD of 7 and 20 is 1, so -7/20 is in simplest form

Final answer:

(3/4) × (-2/5) ÷ (6/7) = -7/20

Applied rules:

• Division to multiplication: a ÷ b = a × (1/b)

• Fraction multiplication: (a/b) × (c/d) = (ac)/(bd)

• Simplification: Divide numerator and denominator by their GCD

3 4 -2 5 6 7 × ÷ → 7 6 = -7 20 Step 1: Convert division to multiplication by reciprocal Step 2: Multiply numerators: 3 × (-2) × 7 = -42 Step 3: Multiply denominators: 4 × 5 × 6 = 120 Step 4: Simplify: -42/120 = -7/20 Fraction Operations: Multiplication and Division
3 Order of Operations (PEMDAS)
Exercise 3
Evaluate: 24 ÷ 4 × (3 + 2) - 5² + 6. Show your work using the order of operations.
Definition:

Order of Operations: PEMDAS/BODMAS - Parentheses, Exponents, Multiplication/Division (left to right), Addition/Subtraction (left to right).

PEMDAS: Parentheses, Exponents, Multiplication, Division, Addition, Subtraction.

Grouping Symbols: Parentheses, brackets, braces that indicate priority.

Step 1: Evaluate expressions inside parentheses first

24 ÷ 4 × (3 + 2) - 5² + 6

= 24 ÷ 4 × 5 - 5² + 6

Step 2: Evaluate exponents

24 ÷ 4 × 5 - 5² + 6

= 24 ÷ 4 × 5 - 25 + 6

Step 3: Perform multiplication and division from left to right

24 ÷ 4 × 5 - 25 + 6

= 6 × 5 - 25 + 6 (division first: 24 ÷ 4 = 6)

= 30 - 25 + 6 (multiplication next: 6 × 5 = 30)

Step 4: Perform addition and subtraction from left to right

30 - 25 + 6

= 5 + 6 (subtraction first: 30 - 25 = 5)

= 11 (addition: 5 + 6 = 11)

Step 5: Verify the calculation

Original: 24 ÷ 4 × (3 + 2) - 5² + 6

= 24 ÷ 4 × 5 - 25 + 6

= 6 × 5 - 25 + 6

= 30 - 25 + 6

= 5 + 6 = 11 ✓

Final answer:

24 ÷ 4 × (3 + 2) - 5² + 6 = 11

Applied rules:

• PEMDAS order: Parentheses → Exponents → Multiplication/Division → Addition/Subtraction

• Left-to-right: Operations of same precedence are evaluated left to right

• Grouping first: Always evaluate inside grouping symbols first

24 ÷ 4 × (3 + 2) - 5² + 6 (3 + 2) = 5 24 ÷ 4 × 5 - 5² + 6 5² = 25 24 ÷ 4 × 5 - 25 + 6 24 ÷ 4 = 6 6 × 5 - 25 + 6 6 × 5 = 30 30 - 25 + 6 = 11 Order of Operations (PEMDAS) Process
Operation Concepts, Rules and Methods
a + b = c, a - b = d, a × b = e, a ÷ b = f
Basic Arithmetic Operations
Commutative Law
a + b = b + a
Order doesn't matter for addition
Associative Law
(a + b) + c = a + (b + c)
Grouping doesn't matter for addition
Distributive Law
a(b + c) = ab + ac
Multiplication distributes over addition
Key definitions:

Real Number: Any number that can be represented on the number line, including rational and irrational numbers.

Operation: A mathematical procedure that combines numbers to produce another number.

Closure Property: The result of an operation on real numbers is always a real number.

Identity Element: A number that leaves others unchanged when used in an operation (0 for addition, 1 for multiplication).

Operation methodology:
  1. Identify operation type: Determine if addition, subtraction, multiplication, or division
  2. Apply sign rules: Handle positive and negative numbers correctly
  3. Follow order of operations: Use PEMDAS/BODMAS when multiple operations exist
  4. Perform calculation: Execute the mathematical operation
  5. Verify result: Check reasonableness and accuracy
Tip 1: When adding numbers with different signs, subtract absolute values and take the sign of the larger.
Tip 2: Multiplying two negative numbers gives a positive result.
Tip 3: Always simplify fractions by finding the GCD of numerator and denominator.
Tip 4: Remember PEMDAS: Parentheses, Exponents, Multiplication/Division, Addition/Subtraction.

Key characteristics: Commutative for addition/multiplication, associative, distributive, closed under operations.
Common applications: Calculations in science, finance, engineering, and everyday problem solving.

Solution: Exercises 4 to 5
4 Operations with Irrational Numbers
Exercise 4
Simplify: 2√3 + 5√3 - √3. Express your answer in simplest radical form.
Definition:

Irrational Number: A real number that cannot be expressed as a ratio of integers.

Like Radicals: Radicals with the same radicand (number under the root sign).

Radical Simplification: Expressing radicals in their simplest form.

Step 1: Identify like radicals

All terms have the same radical: √3

2√3, 5√3, and -√3 are like radicals

Step 2: Combine coefficients

2√3 + 5√3 - √3 = (2 + 5 - 1)√3

= (6)√3

= 6√3

Step 3: Verify the result

2√3 ≈ 2(1.732) = 3.464

5√3 ≈ 5(1.732) = 8.660

√3 ≈ 1.732

3.464 + 8.660 - 1.732 = 10.392

6√3 ≈ 6(1.732) = 10.392 ✓

Final answer:

2√3 + 5√3 - √3 = 6√3

Applied rules:

• Like radical rule: Only radicals with the same radicand can be combined

• Radical addition: a√n + b√n = (a + b)√n

• Coefficient combination: Add/subtract coefficients while keeping the radical

2√3 5√3 -√3 + + = 6√3 2 5 -1 = 6 × √3 Combine coefficients: (2 + 5 - 1) = 6, keep common radical √3 Result: 6√3 Operations with Like Radicals
5 Exponent and Radical Operations
Exercise 5
Evaluate: √(16 × 9) ÷ (2³ - 4) + √25. Show all steps using order of operations.
Definition:

Exponent: A number indicating how many times a base is multiplied by itself.

Radical: An expression involving a root (square root, cube root, etc.).

Order of Operations: PEMDAS/BODMAS rules for evaluating expressions.

Step 1: Evaluate expressions inside radicals and parentheses

√(16 × 9) = √144 = 12

2³ - 4 = 8 - 4 = 4

√25 = 5

Step 2: Substitute simplified values

√(16 × 9) ÷ (2³ - 4) + √25

= 12 ÷ 4 + 5

Step 3: Perform division

12 ÷ 4 + 5 = 3 + 5

Step 4: Perform addition

3 + 5 = 8

Step 5: Verify the result

Original: √(16 × 9) ÷ (2³ - 4) + √25

= √144 ÷ (8 - 4) + 5

= 12 ÷ 4 + 5

= 3 + 5 = 8 ✓

Final answer:

√(16 × 9) ÷ (2³ - 4) + √25 = 8

Applied rules:

• Radical multiplication: √(ab) = √a × √b

• Order of operations: Evaluate exponents and radicals before multiplication/division

• Division: Perform before addition when no grouping symbols

√(16 × 9) ÷ (2³ - 4) + √25 √(16 × 9) = √144 = 12 2³ = 8 8 - 4 = 4 √25 = 5 12 ÷ 4 + 5 12 ÷ 4 = 3 3 + 5 = 8 Order of Operations: Radicals and Exponents

Questions & Answers

Question: Why do we need to follow the order of operations? Can't I just calculate from left to right?

Answer: The order of operations (PEMDAS) ensures everyone interprets expressions the same way:

  • Without rules: 2 + 3 × 4 could equal 20 (left to right) or 14 (order of operations)
  • With PEMDAS: Multiplication comes before addition, so 2 + 3 × 4 = 2 + 12 = 14
  • Consistency: Mathematical communication requires agreed-upon conventions

Think of it like grammar in language - it prevents ambiguity.

The order follows mathematical logic: grouping → exponents → multiplication/division → addition/subtraction.

Always follow PEMDAS: Parentheses, Exponents, Multiplication/Division (left to right), Addition/Subtraction (left to right).

Question: What's the difference between rational and irrational numbers in operations?

Answer: Operations behave differently with rational vs irrational numbers:

  • Rational + Rational = Rational (e.g., 1/2 + 1/3 = 5/6)
  • Rational + Irrational = Irrational (e.g., 2 + √3 is irrational)
  • Irrational + Irrational = Either (e.g., √2 + √3 is irrational, but √2 + (-√2) = 0 is rational)

When multiplying: Rational × Irrational = Irrational (unless rational is 0)

When dividing: Similar rules apply, with extra care for division by zero.

For mixed operations, the result is usually irrational unless terms cancel out.

Question: How do I know if a radical expression is in simplest form?

Answer: A radical is in simplest form when:

  • No perfect square factors: The radicand has no perfect square factors other than 1
  • No fractions under radical: No fractions inside the radical sign
  • No radicals in denominator: Rationalize denominators if needed

Example: √12 = √(4 × 3) = 2√3 (simplified)

Example: √18 = √(9 × 2) = 3√2 (simplified)

Example: √20 = √(4 × 5) = 2√5 (simplified)

Always factor out the largest perfect square from the radicand.

Question: What happens when I have both multiplication and division in an expression?

Answer: Multiplication and division have equal precedence and are performed left to right:

Example: 24 ÷ 4 × 2

Step 1: 24 ÷ 4 = 6 (leftmost operation first)

Step 2: 6 × 2 = 12

Wrong approach: 24 ÷ (4 × 2) = 24 ÷ 8 = 3 (incorrect)

The same rule applies to addition and subtraction.

Remember: PEMDAS means "Parentheses, Exponents, Multiplication/Division (left to right), Addition/Subtraction (left to right)".

Question: How do I multiply fractions with radicals in the numerator or denominator?

Answer: Follow the same rules as regular fraction multiplication, but handle radicals separately:

Rule: (a/b) × (c/d) = (ac)/(bd)

Example: (2/√3) × (3/√5) = (2 × 3)/(√3 × √5) = 6/√15

Then rationalize: 6/√15 × √15/√15 = 6√15/15 = 2√15/5

For radicals: √a × √b = √(ab)

Always rationalize denominators containing radicals in final answers.