Example: √(2 x 8) = √(16) = 4
This is the same as: √2 x √8 = 2 x 2 = 4
(b) The square root of a quotient is equal to the quotient of the square roots:
√(a / b) = √a / √b
Example: √(9 / 4) = √(2.25) = 1.5
This is the same as: √9 / √4 = 3 / 2 = 1.5
(c) The square of a square root eliminates the square root:
(√a)² = a
Example: (√5)² = 5
Example: Simplify √72
Prime factorization of 72 : 2 x 2 x 2 x 3 x 3
Identify pairs: (2 x 2) x 2 x (3 x 3)
Simplify: √72 = √(4 x 2 x 9) = 2 x √2 x 3 = 6√2
Example: Solve √(x + 2) = 3
Step 1: Isolate the square root: √(x + 2) = 3
Step 2: Square both sides: (√(x + 2))² = 3²
Step 3: Solve the new equation: x + 2 = 9
Step 4: Solve for x: x = 7
Examples: Square Roots:
(a) √4 = 2, because 2 × 2 = 4
(b) √81 = 9, because 9 × 9 = 81
(c) √121 = 11, because 11 × 11 = 121
√50 = √(2 * 5²) = 5√2
Q.2. Solve the equation: √(2x  4) = 4
√(2x  4) = 4 → (2x  4) = 16 → 2x = 20 → x = 10
Q.3. If √(x²  4) = 5, what is the value of x?
x²  4 = 25 → x² = 29 → x = ±√29
Q.4. Simplify the expression: 2√18 + 3√8
2√18 + 3√8 = 2(3√2) + 3(2√2) = 6√2 + 6√2 = 12√2
Q.5. Solve the equation: 2√(x + 3)  4 = 6
2√(x + 3)  4 = 6 → 2√(x + 3) = 10 → √(x + 3) = 5 → x + 3 = 25 → x = 22
Q.6. Is 50 a perfect square? If not, find the closest perfect squares and their square roots.
No, 50 is not a perfect square. The closest perfect squares are 49 and 64, and their square roots are 7 and 8, respectively.
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