Square Root and Cube Root— Concepts, Formulas & Shortcuts
- Prime factorise, then take one factor from each pair (square root) or each triple (cube root).
- A perfect square never ends in 2, 3, 7 or 8.
- Squares to memorise: 15² = 225, 20² = 400, 25² = 625, 30² = 900, 34² = 1156.
- Cubes to memorise: 6³ = 216, 7³ = 343, 8³ = 512, 9³ = 729, 10³ = 1000.
- Decimals: √0.09 = 0.3 — halve the number of decimal places for a square root, take a third for a cube root.
- √(a/b) = √a/√b, and √a × √b = √(ab).
Square Root and Cube Root Practice Questions with Answers
Attempt each question first, then open the explanation. All 21 questions below are free to read and require no signup.
Q1.The value of √1156 is:
Moderate- A24
- B28
- C34
- D38
Square Root and Cube Root question 1 of 21+Show Answer & Explanation
Answer: C. 34
Explanation: 34² = 1,156. The last digit 6 also rules out 28 and 38 immediately.
Q2.The cube root of 343 is:
Easy- A6
- B7
- C8
- D9
Square Root and Cube Root question 2 of 21+Show Answer & Explanation
Answer: B. 7
Explanation: 7 × 7 × 7 = 343.
Q3.The value of √0.09 is:
Easy- A0.03
- B0.3
- C3
- D0.9
Square Root and Cube Root question 3 of 21+Show Answer & Explanation
Answer: B. 0.3
Explanation: 0.3 × 0.3 = 0.09; the two decimal places halve to one.
Q4.The value of √(1/16) is:
Easy- A1/2
- B1/4
- C1/8
- D1/16
Square Root and Cube Root question 4 of 21+Show Answer & Explanation
Answer: B. 1/4
Explanation: The square root distributes over a fraction: √1/√16 = 1/4.
Q5.The smallest number by which 1,152 must be multiplied to make it a perfect square is:
Difficult- A2
- B3
- C4
- D6
Square Root and Cube Root question 5 of 21+Show Answer & Explanation
Answer: A. 2
Explanation: 1152 = 2⁷ × 3². The exponent of 2 is odd, so one more 2 makes it 2⁸ × 3², a perfect square.
Q6.The cube root of 0.000216 is:
Difficult- A0.006
- B0.06
- C0.6
- D6
Square Root and Cube Root question 6 of 21+Show Answer & Explanation
Answer: B. 0.06
Explanation: 0.06³ = 0.000216; six decimal places divide by three to give two.
Q7.Given √5 ≈ 2.236, the value of √245 is approximately:
Moderate- A11.18
- B13.42
- C15.65
- D17.89
Square Root and Cube Root question 7 of 21+Show Answer & Explanation
Answer: C. 15.65
Explanation: √245 = √(49 × 5) = 7√5 ≈ 7 × 2.236 = 15.652.
Q8.The value of √625 is:
Easy- A15
- B20
- C25
- D35
Square Root and Cube Root question 8 of 21+Show Answer & Explanation
Answer: C. 25
Explanation: A perfect square ending in 5 has a root ending in 5, and 25 × 25 = 625.
Q9.The cube root of 512 is:
Easy- A6
- B7
- C8
- D9
Square Root and Cube Root question 9 of 21+Show Answer & Explanation
Answer: C. 8
Explanation: 8 × 8 × 8 = 512.
Q10.The value of √0.0004 is:
Moderate- A0.002
- B0.02
- C0.2
- D2
Square Root and Cube Root question 10 of 21+Show Answer & Explanation
Answer: B. 0.02
Explanation: 0.02 × 0.02 = 0.0004; four decimal places halve to two.
Q11.The value of √(144/196) is:
Moderate- A6/7
- B12/14
- C3/7
- D4/7
Square Root and Cube Root question 11 of 21+Show Answer & Explanation
Answer: A. 6/7
Explanation: √144/√196 = 12/14 = 6/7 in lowest terms.
Q12.The smallest number by which 2,187 must be divided to make it a perfect square is:
Difficult- A3
- B7
- C9
- D27
Square Root and Cube Root question 12 of 21+Show Answer & Explanation
Answer: A. 3
Explanation: 2187 = 3⁷. Dividing by 3 gives 3⁶ = 729 = 27², a perfect square.
Q13.The value of ∛(27/64) is:
Moderate- A3/4
- B9/16
- C3/8
- D1/2
Square Root and Cube Root question 13 of 21+Show Answer & Explanation
Answer: A. 3/4
Explanation: The cube root distributes over a fraction: ∛27 = 3 and ∛64 = 4, giving 3/4.
Q14.If √x = 12, then x equals:
Easy- A24
- B36
- C121
- D144
Square Root and Cube Root question 14 of 21+Show Answer & Explanation
Answer: D. 144
Explanation: Squaring both sides: x = 144.
Q15.The value of √256 is:
Easy- A14
- B16
- C18
- D24
Square Root and Cube Root question 15 of 21+Show Answer & Explanation
Answer: B. 16
Explanation: 256 is a perfect square because 16 × 16 = 256.
Q16.The value of √441 is:
Moderate- A19
- B21
- C23
- D29
Square Root and Cube Root question 16 of 21+Show Answer & Explanation
Answer: B. 21
Explanation: 441 ends in 1, so its root ends in 1 or 9; testing gives 21 × 21 = 441.
Q17.The cube root of 216 is:
Easy- A4
- B5
- C6
- D8
Square Root and Cube Root question 17 of 21+Show Answer & Explanation
Answer: C. 6
Explanation: 6 × 6 × 6 = 216.
Q18.The value of √0.25 is:
Easy- A0.05
- B0.5
- C2.5
- D5
Square Root and Cube Root question 18 of 21+Show Answer & Explanation
Answer: B. 0.5
Explanation: 0.5 × 0.5 = 0.25.
Q19.The cube root of 1,000 is:
Easy- A10
- B20
- C30
- D100
Square Root and Cube Root question 19 of 21+Show Answer & Explanation
Answer: A. 10
Explanation: The cube root undoes the cube, and 10 × 10 × 10 = 1,000.
Q20.The value of √(49 × 81) is:
Moderate- A56
- B63
- C72
- D81
Square Root and Cube Root question 20 of 21+Show Answer & Explanation
Answer: B. 63
Explanation: √49 × √81 = 7 × 9 = 63.
Q21.If x² = 169, then x (taking the positive root) is:
Easy- A11
- B12
- C13
- D14
Square Root and Cube Root question 21 of 21+Show Answer & Explanation
Answer: C. 13
Explanation: Take the square root of both sides: √169 = 13.
Square Root and Cube Root — Frequently Asked Questions
How do I find the smallest number to multiply by to make a perfect square?+
Prime factorise and look for factors appearing an odd number of times. 1152 = 2⁷ × 3², where 2 appears seven times, so multiplying by one more 2 makes every exponent even.
How do decimal places behave under a square root?+
The number of decimal places halves. √0.09 = 0.3 (two places become one) and √0.0016 = 0.04.
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