Exercise practice 2: A Square and a Cube

4.1 Exercise practice 2: Figure it Out, page 16

The second Figure it Out section deals with cubes and cube roots. Full statements and reasoning are below.

4.1.1 E2.1 Two cube roots

Question. Take the cube root of 27000 and of 10648.

Answer. The first is 30; the second is 22.

Key steps. For 27000: 27000 = 27 x 1000 = 3^3 x 10^3 = (3 x 10)^3 = 30^3. For 10648: divide by 8 to get 1331 = 11^3, so 10648 = 2^3 x 11^3 = (2 x 11)^3 = 22^3. Check: 30^3 = 27000 and 22^3 = 10648.

4.1.2 E2.2 Multiplying 1323 to make a cube

Question. What number should 1323 be multiplied by to make it a cube number?

Answer. Multiply by 7; the product is 9261 = 21^3.

Key steps. Factorise 1323 = 3^3 x 7^2. The 3s already form a triplet, but the two 7s need one more 7 to make a triplet. Multiplying by 7 gives 3^3 x 7^3 = (3 x 7)^3 = 21^3 = 9261. So the smallest multiplier is 7.

4.1.3 E2.3 True or false with reasons

Question. Test five claims and give a reason for each: the cube of an odd number is even; no perfect cube ends in 8; a two-digit number can cube to a three-digit number; a two-digit number can cube to seven or more digits; every cube has an odd number of factors.

Answer. All five claims are false.

Key steps. Odd numbers cubed stay odd, for example 3^3 = 27, so the first claim fails. The second fails because 2^3 = 8 ends in 8. The third fails because the smallest two-digit base is 10, and 10^3 = 1000 already has four digits. The fourth fails because the largest two-digit base is 99, and 99^3 = 970299 has only six digits. The fifth fails because a cube can have an even factor count: the number 8 has the four factors 1, 2, 4 and 8.

4.1.4 E2.4 Guessing cube roots by last digits

Question. Since 1331 is a perfect cube, estimate its cube root without factorising. Then estimate the roots of 4913, 12167 and 32768 the same way.

Answer. 11, 17, 23 and 32 respectively.

Key steps. Use the last digit of the cube. A cube ending in 1 comes from a root ending in 1, so 1331 has root 11. A cube ending in 3 comes from a root ending in 7, and 4913 lies between 10^3 = 1000 and 20^3 = 8000, so test 17: 17^3 = 4913. A cube ending in 7 comes from a root ending in 3; 12167 lies between 20^3 and 30^3, and 23^3 = 12167. A cube ending in 8 comes from a root ending in 2; 32768 lies between 30^3 = 27000 and 40^3 = 64000, and 32^3 = 32768.

4.1.5 E2.5 Greatest difference

Question. Which is greatest: (i) 67^3 - 66^3, (ii) 43^3 - 42^3, (iii) 67^2 - 66^2, (iv) 43^2 - 42^2? Explain.

Answer. (i) 67^3 - 66^3 is the greatest.

Key steps. For consecutive numbers a and b = a - 1, a^3 - b^3 = a^2 + ab + b^2, while a^2 - b^2 = a + b. Compute: 67^3 - 66^3 = 4489 + 4422 + 4356 = 13267; 43^3 - 42^3 = 1849 + 1806 + 1764 = 5419; 67^2 - 66^2 = 133; 43^2 - 42^2 = 85. So (i) is much larger than the others. The larger the base, the larger the difference of cubes, and cube differences grow faster than square differences.

4.2 Checking every answer

Cube roots can be checked by cubing the answer. If the cube root of 27000 is claimed to be 30, then 30^3 must equal 27000: 30 x 30 = 900 and 900 x 30 = 27000, so the claim is correct. The same check works for 10648: 22 x 22 = 484, 484 x 22 = 10648.

For the true or false questions, one counterexample is enough to prove a statement false. The number 8 ends in 8 and is a cube, which settles statement (ii). The number 8 has four factors, which settles statement (v). A single genuine example overturns a claim that says something is always true.

For the last digit method, remember that the cube root of a number ending in 3 ends in 7 and the cube root of a number ending in 7 ends in 3. This crossing of digits is easy to mix up, so write the pair down before choosing the root.

A fourth check is to estimate the size. The cube of 20 is 8000 and the cube of 30 is 27000. Any cube root between 20 and 30 must belong to a number in that range, so a root ending in 2 for 32768 must be 22 or 32; since 32768 is above 27000, it is 32. Size and last digit together pin the root down. Comparing the digit count with the nearby cube also catches a wrong choice. Check the digit count as well.