A Square and A Cube — Practice

Question 1

What is 15²?

Accepted Answer

✦ Answer: 15² = 15 × 15 = 225. Memorise this — it's one of the squares to know up to 30².

Question 2

What is 6³?

Accepted Answer

✦ Answer: 6³ = 6 × 6 × 6 = 216. Memorise cubes up to 15³ for quick recall.

Question 3

Find √2304 using prime factorisation.

Accepted Answer

Step 1 — Prime factorise 2304.
  2304 = 2 × 1152 = 2 × 2 × 576
       = 2 × 2 × 2 × 288
       = 2 × 2 × 2 × 2 × 144
       = 2 × 2 × 2 × 2 × 2 × 72
       = 2 × 2 × 2 × 2 × 2 × 2 × 36
       = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 18
       = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 9
       = 2⁸ × 3²

Step 2 — Pair the factors.
  2304 = (2 × 2) × (2 × 2) × (2 × 2) × (2 × 2) × (3 × 3)

Step 3 — Take one from each pair.
  2 × 2 × 2 × 2 × 3 = 48

Step 4 — Verify.
  48² = 48 × 48 = 2304 ✓

✦ Answer: √2304 = 48.

Question 4

Find the cube root of 2197 by prime factorisation.

Accepted Answer

Step 1 — Prime factorise 2197.
  2197 ÷ 13 = 169
  169 ÷ 13 = 13
  13 ÷ 13 = 1
  So 2197 = 13 × 13 × 13 = 13³

Step 2 — Group in triplets.
  2197 = (13 × 13 × 13)
  One complete triplet of 13s.

Step 3 — Take one from each triplet.
  Only one triplet, so take 13.

Step 4 — Verify.
  13³ = 13 × 13 × 13 = 169 × 13 = 2197 ✓

✦ Answer: ∛2197 = 13.

Question 5

Find the smallest number by which 392 must be MULTIPLIED to make it a perfect cube, and find the cube root of the resulting number.

Accepted Answer

Step 1 — Prime factorise 392.
  392 = 2 × 196
      = 2 × 2 × 98
      = 2 × 2 × 2 × 49
      = 2 × 2 × 2 × 7 × 7
      = 2³ × 7²

Step 2 — Check for triplets.
  2³ — a complete triplet ✓
  7² — only two 7s, need one more 7 to make a triplet

Step 3 — Find the smallest multiplier.
  To complete the triplet of 7s, multiply by 7.
  Smallest number to multiply = 7.

Step 4 — Find the perfect cube.
  392 × 7 = 2744
  Let's verify: 2744 = 2³ × 7³ = (2 × 7)³ = 14³

Step 5 — Find cube root.
  ∛2744 = ∛(14³) = 14

Step 6 — Verify.
  14³ = 14 × 14 × 14 = 196 × 14 = 2744 ✓

✦ Answer: Smallest multiplier = 7. The perfect cube formed = 2744. Its cube root = 14.

Question 6

Which of these numbers is a perfect square?

Accepted Answer

1024

Question 7

A number ending in which digit can NEVER be a perfect square?

Accepted Answer

8

Question 8

The smallest number by which 72 must be multiplied to make it a perfect square is:

Accepted Answer

2

Question 9

Which of the following is a perfect cube?

Accepted Answer

216

Question 10

Find √784 using prime factorisation.

Accepted Answer

28

Question 11

Find the cube root of 3375 by prime factorisation.

Accepted Answer

15

Question 12

Find the smallest number by which 2028 must be divided so that the quotient is a perfect square. Also find the square root of the quotient.

Accepted Answer

3; 26

Practice — A Square and A Cube

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