Math Magic: Square Root Explorer (3.7)

🔍 Problem 1: Find the square root of rational expressions

(i) Find √(400x⁴y¹²z¹⁶ / 100x⁸y⁴z⁴)

Step 1: Simplify the expression inside the square root:

√(400x⁴y¹²z¹⁶ / 100x⁸y⁴z⁴) = √(4x⁴⁻⁸y¹²⁻⁴z¹⁶⁻⁴)

Step 2: Simplify exponents:

= √(4x⁻⁴y⁸z¹²)

Step 3: Take square root of each term:

= √4 × √x⁻⁴ × √y⁸ × √z¹²

Step 4: Calculate each square root:

= 2 × x⁻² × y⁴ × z⁶

✓

Final Answer:

= 2y⁴z⁶ / x²

(ii) Find √[(7x² + 2√14x + 2) / (x² - ½x + 1/16)]

Step 1: Factor numerator and denominator separately.

Step 2: Numerator: 7x² + 2√14x + 2 = (√7x + √2)²

Because (a + b)² = a² + 2ab + b² where a = √7x, b = √2

Step 3: Denominator: x² - ½x + 1/16 = (x - ¼)²

Perfect square pattern: (a - b)² = a² - 2ab + b²

Step 4: Now the expression becomes:

√[(√7x + √2)² / (x - ¼)²]

✓

Final Answer:

= (√7x + √2) / (x - ¼)

(iii) Find √[121(a+b)⁸(x+y)⁸(b-c)⁸ / 81(b-c)⁴(a-b)¹²(b-c)⁴]

Step 1: Combine like terms in denominator:

Denominator: 81(b-c)⁴⁺⁴(a-b)¹² = 81(b-c)⁸(a-b)¹²

Step 2: Rewrite the expression:

√[121(a+b)⁸(x+y)⁸(b-c)⁸ / 81(a-b)¹²(b-c)⁸]

Step 3: Cancel (b-c)⁸ terms:

= √[121(a+b)⁸(x+y)⁸ / 81(a-b)¹²]

Step 4: Take square root of each term:

= 11(a+b)⁴(x+y)⁴ / 9(a-b)⁶

✓

Final Answer:

= 11(a+b)⁴(x+y)⁴ / 9(a-b)⁶

🔢 Problem 2: Find the square root of expressions

(i) Find √(4x² + 20x + 25)

Step 1: Recognize perfect square trinomial pattern:

a² + 2ab + b² = (a + b)²

Step 2: Identify terms:

4x² = (2x)², 25 = 5², 20x = 2 × 2x × 5

Step 3: Rewrite as perfect square:

4x² + 20x + 25 = (2x + 5)²

✓

Final Answer:

√(4x² + 20x + 25) = 2x + 5

(ii) Find √(9x² - 24xy + 30xz - 40yz + 25z² + 16y²)

Step 1: Group terms to find perfect square pattern.

Step 2: Rearrange terms:

9x² - 24xy + 16y² + 30xz - 40yz + 25z²

Step 3: First three terms form perfect square:

9x² - 24xy + 16y² = (3x - 4y)²

Step 4: Last three terms have common factor:

30xz - 40yz + 25z² = 5z(6x - 8y + 5z)

Step 5: Notice 6x-8y is 2(3x-4y):

= 5z[2(3x-4y) + 5z] = 10z(3x-4y) + 25z²

✓

Final Answer:

The expression is (3x - 4y + 5z)², so √... = 3x - 4y + 5z

(iii) Find √[(4x² - 9x + 2)(7x² - 13x - 2)(28x² - 3x - 1)]

Step 1: Factor each quadratic expression.

Step 2: Factor first expression:

4x² - 9x + 2 = (4x - 1)(x - 2)

Step 3: Factor second expression:

7x² - 13x - 2 = (7x + 1)(x - 2)

Step 4: Factor third expression:

28x² - 3x - 1 = (7x + 1)(4x - 1)

Step 5: Multiply them together:

(4x-1)(x-2) × (7x+1)(x-2) × (7x+1)(4x-1)

Step 6: Rearrange terms:

= [(4x-1)(7x+1)(x-2)]²

✓

Final Answer:

√... = (4x - 1)(7x + 1)(x - 2)

(iv) Find √[(2x² + 17/6x + 1)(3/2x² + 4x + 2)(4/3x² + 11/3x + 2)]

Step 1: Multiply each expression by appropriate factor to eliminate fractions.

Step 2: First expression ×6:

12x² + 17x + 6

Step 3: Second expression ×2:

3x² + 8x + 4

Step 4: Third expression ×3:

4x² + 11x + 6

Step 5: Factor each:

(3x+2)(4x+3), (3x+2)(x+2), (x+2)(4x+3)

Step 6: Multiply them:

(3x+2)²(4x+3)²(x+2)²

✓

Final Answer:

√... = (3x + 2)(4x + 3)(x + 2) / √(6×2×3) = (3x+2)(4x+3)(x+2)/6