Cube Numbers

Worked example

1. 1

   $4^3=4\times 4\times 4$ — The small 3 says how many fours to multiply. Write out all three fours so none gets lost.

2. 2

   $4\times 4=16$ — Multiply the first two fours. You multiply two numbers at a time.

3. 3

   $16\times 4=64$ — Now multiply 16 by the third four. All three fours are used, so $4^3=64$.

Worked example

1. 1

   $13^3=13\times 13\times 13$ — Write out three thirteens, because the small 3 means three of them multiplied.

2. 2

   $13\times 13=169$ — Multiply the first two. This is only $13^2$, the square, so you are not finished.

3. 3

   $169\times 13=2197$ — Multiply by the third 13. All three are used now, so $13^3=2197$.

Worked example

1. 1

   $(-3{)}^3=(-3)\times (-3)\times (-3)$ — Keep −3 in brackets, so the minus is cubed too.

2. 2

   $(-3)\times (-3)=9$ — The first two: minus times minus is plus, so +9.

3. 3

   $9\times (-3)=-27$ — Plus times minus is minus. So a negative number cubed stays negative.

Worked example

1. 1

   $0.2^3=0.2\times 0.2\times 0.2$ — Three lots of 0.2 multiplied, the same as any cube.

2. 2

   $2\times 2\times 2=8$ — Ignore the decimal points for now and multiply the digits.

3. 3

   $0.2^3=0.008$ — Each 0.2 has one decimal place, so the answer has 1 + 1 + 1 = 3 decimal places.

Worked example

1. 1

   ${\left(\frac{2}{5}\right)}^3=\frac{2^3}{5^3}$ — The power acts on the whole fraction, so cube the top and the bottom.

2. 2

   $=\frac{8}{125}$ — 2 × 2 × 2 = 8 and 5 × 5 × 5 = 125.

Worked example

1. 1

   $1,8,27,64$ — Write the cubes $1^3,2^3,3^3,4^3$ in order. Stop at 64, because it is already bigger than 49, the largest number in the list.

2. 2

   $27$ — Look for a match. 27 is in the list and no other cube is, so the answer is 27.

3. 3

   $3\times 3\times 3=9\times 3=27$ — Check it: 3 × 3 = 9, then 9 × 3 = 27. So 27 really is 3 cubed.

Worked example

1. 1

   $4^3=64,5^3=125,6^3=216$ — Go up the cubes until you pass 150.

2. 2

   $100<125<150$ — Only 125 lands inside the range, so the answer is 125.

Worked example

1. 1

   $600=2^3\times 3\times 5^2$ — Write 600 as a product of primes first. Every prime must end with a power of 3, 6, 9 and so on.

2. 2

   $2^3$ — Three 2s: a full group, so the 2s need nothing.

3. 3

   $3\to 3^3$ — Only one 3, so it needs two more 3s: $3^2$.

4. 4

   $5^2\to 5^3$ — Two 5s, so it needs one more 5.

5. 5

   $3^2\times 5=9\times 5=45$ — Multiply only the missing primes. The smallest number is 45.

6. 6

   $600\times 45=27000=2^3\times 3^3\times 5^3=30^3$ — Check: every power is now 3, and $2\times 3\times 5=30$, so 27000 is 30 cubed.