Test: Square Root And Cube Root- 1 - Bank Exams MCQ

Given:
x / √512 = √648 / x

Cross-multiply:
x² = √512 · √648 = √(512 · 648)

Compute inside the radical:
512 = 2⁹, 648 = 2³ · 3⁴
512 · 648 = 2¹² · 3⁴

So:
x² = √(2¹² · 3⁴)
= 2⁶ · 3²
= 64 · 9
= 576

Hence:
x = ±√576 = ±24

1. Set up digit pairs
   5 . 47 56

2. Integer part
   – Largest square ≤ 5 is 2²=4.
   – Root so far: 2. Remainder: 5−4=1.

3. First decimal
   – Bring down “47” → 147.
   – Double root “2” → 4_. Find x with (40+x)·x ≤147.
   • x=3 → 43·3=129 ≤147
   • x=4 → 44·4=176 >147
   – Choose x=3. Root: 2.3. Remainder: 147−129=18.

4. Second decimal
   – Bring down “56” → 1856.
   – Double root “23” → 46_. Find y with (460+y)·y ≤1856.
   • y=4 → 464·4=1856 exactly
   – Choose y=4. Root: 2.34. Remainder: 0.

Answer: 2.34 (option d).

To find the cube root of 0.000729, we recognize that 0.000729 can be expressed as (0.09)³.
Therefore, the cube root of 0.000729 is 0.09, which confirms that option 1 (0.09), is the correct answer.

= (12/11) x (11/15) x (15/14)

= 12/14

= 0.85

► Using identity, (a-b)² = a² + b² - 2ab

► Here a = √7 and b = 1/√7

 
= (√7)² + (1/√7)² - 2. √7.1/√7

= 7 + 1/7 - 2

= 5 + 1/7

= 36/7

1. Use last-digit rule

• Perfect squares end with: 0,1,4,5,6,9.

• 16641 ends with 1 → root ends with 1 or 9.

2. Use digit-sum (9’s test) for a quick filter

• Sum of digits = 1+6+6+4+1 = 18 → 18 ≡ 0 (mod 9) → number is a multiple of 9.

• A perfect square that’s a multiple of 9 must have a root that’s a multiple of 3.

• Among endings {1,9}, only 9 is a multiple of 3 → root ends with 9.

3. Find the two bounding tens

• 120² = 14400, 130² = 16900
  → √16641 is between 120 and 130.

• With last digit 9 and within 120–130, the only candidate is 129.

4. Quick confirm using (a−b)² trick

• 129² = (130−1)² = 130² − 2·130 + 1 = 16900 − 260 + 1 = 16641.

Squaring both side,
⇒ 0.0576 x a = 0.0576
⇒ a = 1

Squaring both side,

⇒ 0.000256 x a = 2.56

⇒ a = 2.56/0.000256

⇒ a = 10000

Let the integers be n, n+1, n+2.
n(n+1)(n+2) = 15600 → try n = 24 since 24·25·26 = 15600. So the integers are 24, 25, 26.

Sum of squares = 24² + 25² + 26²
= 576 + 625 + 676
= 1877.

Answer: 1877.