Solved Exercises on Square Roots and Cube Roots in Grade 8

Master square roots and cube roots: properties, operations, and real-world applications through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Perfect Squares and Cubes
Exercise 1
Find the square root of 144 and the cube root of 216. Verify your answers by squaring and cubing respectively.
Definition:

Square Root: If x² = a, then x = √a (the positive root is called the principal square root).

Cube Root: If x³ = a, then x = ∛a (there is only one real cube root).

Perfect Square: A number that is the square of an integer.

Perfect Cube: A number that is the cube of an integer.

Method for finding roots:
  1. Factor the number into prime factors
  2. For square roots: Group factors in pairs
  3. For cube roots: Group factors in triplets
  4. Take one factor from each group
  5. Verify by exponentiation
Step 1: Find √144

Factor 144: 144 = 12 × 12 = 2² × 3 × 2² × 3 = 2⁴ × 3²

Group in pairs: (2²)(2²)(3²)

Take one from each pair: 2 × 2 × 3 = 12

Therefore, √144 = 12

Step 2: Find ∛216

Factor 216: 216 = 6 × 6 × 6 = 216

Or: 216 = 2³ × 3³ = (2 × 3)³ = 6³

Take one from each triplet: 2 × 3 = 6

Therefore, ∛216 = 6

Step 3: Verify √144 = 12

12² = 12 × 12 = 144 ✓

Step 4: Verify ∛216 = 6

6³ = 6 × 6 × 6 = 36 × 6 = 216 ✓

Final answer:

√144 = 12 and ∛216 = 6.

Applied rules:

• Square root property: If x² = a, then x = ±√a (but principal root is positive)

• Cube root property: If x³ = a, then x = ∛a (unique real root)

• Perfect squares: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, ...

12² = 144 √144 = 12 6³ = 216 ∛216 = 6 12 × 12 = 144 6 × 6 × 6 = 216 Perfect Squares and Cubes
2 Operations with Radicals
Exercise 2
Simplify: 2√18 + 3√2 - √32. Express your answer in simplest radical form.
Definition:

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

Simplest Radical Form: When the radicand has no perfect square factors other than 1.

Rationalizing: Removing radicals from denominators.

Step 1: Simplify each radical

√18 = √(9 × 2) = √9 × √2 = 3√2

√32 = √(16 × 2) = √16 × √2 = 4√2

Step 2: Substitute simplified forms

2√18 + 3√2 - √32

= 2(3√2) + 3√2 - 4√2

= 6√2 + 3√2 - 4√2

Step 3: Combine like terms

Since all terms have √2, we can add coefficients:

(6 + 3 - 4)√2 = 5√2

Step 4: Verify the answer

Original: 2√18 + 3√2 - √32 ≈ 2(4.24) + 3(1.41) - 5.66 ≈ 8.48 + 4.23 - 5.66 ≈ 7.05

Simplified: 5√2 ≈ 5(1.41) ≈ 7.05 ✓

Step 5: Check if in simplest form

5√2: The radicand 2 has no perfect square factors other than 1

Therefore, 5√2 is in simplest form

Final answer:

2√18 + 3√2 - √32 = 5√2

Applied rules:

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

• Like terms: Only radicals with same radicand can be combined

• Distributive property: a√b + c√b = (a + c)√b

2√18 = 2√(9×2) = 2×3√2 = 6√2 √32 = √(16×2) = 4√2 6√2 + 3√2 - 4√2 = (6+3-4)√2 = 5√2 5 like radicals 5√2 Operations with Radicals
3 Real-World Application
Exercise 3
A cube-shaped container has a volume of 729 cubic centimeters. What is the length of one edge of the cube? If the cube is painted on all faces, what is the total surface area?
Definition:

Volume of Cube: V = s³ where s is the length of an edge.

Surface Area of Cube: SA = 6s² where s is the length of an edge.

Cube Root Application: Finding the edge length when volume is known.

Step 1: Find edge length using cube root

V = s³ = 729

s = ∛729

Step 2: Calculate cube root of 729

Factor 729: 729 = 9³ = (3²)³ = 3⁶ = (3²)³ = 9³

Therefore, ∛729 = 9

Step 3: Calculate surface area

SA = 6s² = 6(9)² = 6 × 81 = 486 square centimeters

Step 4: Verify the calculations

Edge length: 9 cm

Volume check: 9³ = 729 cm³ ✓

Surface area: 6 faces × (9 × 9) = 6 × 81 = 486 cm² ✓

Step 5: Present final answer

Edge length = 9 cm, Surface area = 486 cm²

Final answer:

The edge length is 9 cm and the surface area is 486 square centimeters.

Applied rules:

• Cube volume formula: V = s³

• Cube surface area: SA = 6s²

• Cube root inverse: Cube root reverses cubing operation

9 cm V = s³ 729 = s³ s = ∛729 = 9 SA = 6s² = 6(9²) = 6(81) = 486 Cube Volume and Surface Area
Root Concepts, Rules and Methods
\sqrt[n]{a^n} = |a| \text{ if n is even}, a \text{ if n is odd}
nth Root Property
Square Root
√a = a^(1/2)
√(ab) = √a × √b
Cube Root
∛a = a^(1/3)
∛(abc) = ∛a × ∛b × ∛c
Perfect Square
n² for integer n
√(n²) = |n|
Key definitions:

Principal Square Root: The non-negative number whose square equals the original number.

Radical Expression: An expression containing a root symbol (√, ∛, etc.).

Radicand: The number or expression under the radical sign.

Index: The small number in the radical that indicates the type of root (2 for square root, 3 for cube root).

Root calculation methodology:
  1. Identify perfect squares/cubes: Know the basic ones (1-20)
  2. Prime factorization: Factor the number completely
  3. Group factors: Pairs for square roots, triplets for cube roots
  4. Extract roots: Take one factor per group
  5. Combine results: Multiply extracted factors
Tip 1: Memorize perfect squares up to 20² and perfect cubes up to 10³.
Tip 2: Only combine radicals with the same radicand.
Tip 3: For even roots, consider absolute value for real solutions.
Tip 4: Always verify your answer by raising to the original power.

Key characteristics: Square roots yield positive values (principal root), cube roots yield unique real values.
Common applications: Geometry, physics, engineering, finance, and statistical calculations.
Solution: Exercises 4 to 5
4 Rationalizing Denominators
Exercise 4
Rationalize the denominator of: 5/(√3 + √2). Express your answer in simplest form.
Definition:

Rationalizing Denominator: Eliminating radicals from the denominator by multiplying by the conjugate.

Conjugate: For (a + b), the conjugate is (a - b).

Difference of Squares: (a + b)(a - b) = a² - b².

Step 1: Identify the conjugate

Denominator: √3 + √2

Conjugate: √3 - √2

Step 2: Multiply numerator and denominator by conjugate

5/(√3 + √2) × (√3 - √2)/(√3 - √2)

= 5(√3 - √2) / [(√3 + √2)(√3 - √2)]

Step 3: Simplify the denominator

(√3 + √2)(√3 - √2) = (√3)² - (√2)² = 3 - 2 = 1

Step 4: Simplify the expression

5(√3 - √2) / 1 = 5(√3 - √2) = 5√3 - 5√2

Step 5: Verify the result

Original: 5/(√3 + √2) ≈ 5/(1.732 + 1.414) ≈ 5/3.146 ≈ 1.59

Simplified: 5√3 - 5√2 ≈ 5(1.732) - 5(1.414) ≈ 8.66 - 7.07 ≈ 1.59 ✓

Final answer:

5/(√3 + √2) = 5√3 - 5√2

Applied rules:

• Conjugate multiplication: Eliminates radicals from denominator

• Difference of squares: (a + b)(a - b) = a² - b²

• Rationalization: Required for standard mathematical form

5 √3 + √2 × √3 - √2 √3 - √2 = 5(√3 - √2) 3 - 2 5√3 - 5√2 Denominator: (√3 + √2)(√3 - √2) = 3 - 2 = 1 Rationalizing the Denominator
5 Advanced Root Operations
Exercise 5
If √x = 5 and ∛y = 3, find the value of x + y. Then find √(x + y).
Definition:

Inverse Operations: Square root and square are inverse operations; cube root and cube are inverse operations.

Algebraic Manipulation: Using inverse operations to solve for variables.

Step 1: Find x from √x = 5

To find x, square both sides: (√x)² = 5²

Therefore: x = 25

Step 2: Find y from ∛y = 3

To find y, cube both sides: (∛y)³ = 3³

Therefore: y = 27

Step 3: Calculate x + y

x + y = 25 + 27 = 52

Step 4: Find √(x + y)

√(x + y) = √52

Factor 52: 52 = 4 × 13 = 2² × 13

√52 = √(2² × 13) = 2√13

Step 5: Verify the results

√25 = 5 ✓, ∛27 = 3 ✓

√52 = √(4 × 13) = 2√13 ≈ 2(3.606) ≈ 7.21 ✓

Final answer:

x = 25, y = 27, x + y = 52, and √(x + y) = 2√13.

Applied rules:

• Inverse operations: (√a)² = a, (∛a)³ = a

• Radical simplification: Factor out perfect squares/cubes

• Algebraic manipulation: Apply inverse operations to solve equations

√x = 5 ∛y = 3 x = 25 y = 27 x + y = 52 √(x+y) = 2√13 Advanced Root Operations
Root Theory: Laws, Methods, Definitions, and Formulas
\sqrt{a} \times \sqrt{b} = \sqrt{ab}
Radical Multiplication
Key definitions:

nth Root: If xⁿ = a, then x = √[n]{a}. For square roots, n = 2 (usually omitted).

Principal Root: The positive root when dealing with even indices.

Rational Exponent: √[n]{a} = a^(1/n).

Perfect Powers: Numbers that are exact powers of integers.

Root calculation methodology:
  1. Prime factorization: Factor the radicand completely
  2. Group factors: Group into sets matching the index
  3. Extract roots: Take one factor per group
  4. Handle remainders: Leftover factors stay under the radical
  5. Verify: Check by raising to the original power
Tip 1: For even roots, the radicand must be non-negative for real solutions.
Tip 2: For odd roots, the radicand can be any real number.
Tip 3: Always check if the radicand has perfect square/cube factors.
Tip 4: Rationalize denominators to eliminate radicals from the bottom of fractions.

Key characteristics: Even roots yield positive values (principal root), odd roots preserve sign of radicand.
Common applications: Geometry, physics, engineering, finance, and scientific calculations.
Essential formulas and rules:

• Product rule: √(ab) = √a × √b

• Quotient rule: √(a/b) = √a / √b

• Power rule: (√a)² = a (for a ≥ 0)

• Conjugate rule: (a + √b)(a - √b) = a² - b

• Even index: √[n]{a} requires a ≥ 0 for real solutions

Questions & Answers

Question: Why do we need to rationalize denominators? Isn't it easier to leave them as they are?

Answer: Rationalizing denominators serves several important purposes:

  • Standardization: Creates a consistent form for mathematical expressions
  • Historical reasons: Before calculators, it was easier to divide by rational numbers
  • Further calculations: Simplifies subsequent operations and comparisons
  • Calculus: Necessary for certain limit and derivative calculations

Example: 1/√2 vs √2/2 - the second form is more useful for adding to other expressions.

While leaving radicals in denominators isn't mathematically wrong, rationalizing is considered standard practice.

It's particularly important in higher mathematics like calculus and analysis.

Question: What's the difference between √x² and (√x)²? Aren't they the same thing?

Answer: These expressions are different:

  • √x²: This equals |x| (absolute value of x), defined for all real x
  • (√x)²: This equals x, but only defined for x ≥ 0

Example with x = -3:

√x² = √(-3)² = √9 = 3 = |−3|

(√x)² = (√−3)² is undefined in real numbers!

Example with x = 4:

√x² = √16 = 4 = |4|

(√x)² = (√4)² = 2² = 4

They're equal only when x ≥ 0, but √x² always gives a non-negative result.

Question: Can square roots of negative numbers exist? What about cube roots?

Answer: The behavior differs for even and odd roots:

  • Even roots (square, 4th, 6th, etc.): Cannot have negative radicands in real numbers
  • Odd roots (cube, 5th, 7th, etc.): Can have negative radicands

Square roots: √(-4) is undefined in real numbers (requires imaginary numbers)

Cube roots: ∛(-8) = -2 because (-2)³ = -8

This is because:

Even powers always produce non-negative results (x² ≥ 0)

Odd powers preserve the sign of the base (x³ has same sign as x)

So only odd roots can "undo" negative results.

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

Answer: A radical expression is in simplest form when:

  • No perfect square factors: The radicand has no perfect square factors other than 1
  • No fractions under radical: Radicand is not a fraction
  • No radicals in denominator: Denominator is rationalized

Example of simplest form: √7 - no perfect square factors in 7

Not simplest form: √12 = √(4×3) = 2√3 (12 has perfect square factor 4)

Not simplest form: √(3/4) = √3/2 (radicand is a fraction)

Not simplest form: 1/√2 = √2/2 (radical in denominator)

Always check for perfect square/cube factors in the radicand first.

Question: Where do square roots and cube roots appear in real life?

Answer: Roots appear frequently in real-world applications:

  • Square roots: Calculating distances, areas, standard deviations, and quadratic formulas
  • Cube roots: Finding dimensions of cubes, calculating interest rates, and solving cubic equations
  • Geometry: Pythagorean theorem, circle formulas, volume calculations
  • Physics: Wave equations, oscillations, and acceleration formulas

Examples:

Finding the side length of a square room given its area: √area

Calculating the time for an object to fall: √(2h/g)

Determining the radius of a sphere given its volume: ∛(3V/4π)

In finance, compound interest calculations often involve roots.