Cubes and Cube Roots
Numbers · Cubes and Cube Roots
Syllabus tag: Kenya CBC | Grade 9 Mathematics | Strand 1.0 Numbers | Sub-Strand 1.2 Cubes and Cube Roots (5 Lessons)
Lesson objectives
By the end of this sub-strand, you should be able to:
- Work out the cube of a number in different situations.
- Determine the cube root of a number by factorisation and by using mathematical tables.
- Distinguish clearly between squaring and cubing, and between square roots and cube roots.
- Apply cubes and cube roots to practical problems involving volume and capacity.
Cubes and Cube Roots
A water tank measuring 4 metres each way holds 4 × 4 × 4 cubic metres. That repeated multiplication is what cubing means.
Cube roots run the same journey backwards. Given the space inside, they tell you the length of a side.
a) Why it is called cubing
A cube has the same measurement in all three directions.
To fill it you multiply length by width by height. When all three are equal, you are multiplying a number by itself three times.
That is why 4 × 4 × 4 is written 4³ and read as "four cubed".
b) Cubing a number
Cubing is not the same as multiplying by 3. 4³ is 64, not 12. This is the mistake to watch for.
The sign follows the base. A positive number cubed stays positive. A negative number cubed stays negative, because three minuses multiplied give a minus.
That differs from squaring, where a negative becomes positive.
c) Perfect cubes worth knowing
Some numbers come up again and again. 1, 8, 27, 64, 125, 216, 343, 512, 729 and 1000 are the cubes of 1 to 10.
Learning these ten makes most questions quick, because you will often recognise the answer before calculating.
d) Finding a cube root by factorisation
The cube root of a number is the value that was cubed to make it.
Break the number into prime factors, then look for groups of three identical factors.
Each group of three contributes one factor to the answer. Two groups here, so two factors, and multiplying them gives 6.
Always check by cubing your answer.
Sometimes a factor will not fall into a complete group of three. Then the number is not a perfect cube, and you need tables or a calculator.
e) Using tables and a calculator
For numbers that are not perfect cubes, use the cube root key on a calculator. You can also read the value from mathematical tables.
Estimate first so you can spot a wrong answer. The cube root of 500 lies between 7 and 8. That is because 7³ is 343 and 8³ is 512.
f) Where this is used
A tank builder works back from a required volume to the side length. A packaging designer sizing a cubic carton does the same. Engineers use cube roots when scaling models up to full size.
Words to know
- Cube of a number -- the result of using the number as a factor three times, written with a raised 3.
- Cube root -- the number which, when cubed, gives the original number.
- Perfect cube -- a number whose cube root is a whole number, such as 8, 27, 64, 125 and 216.
- Prime factorisation -- expressing a number as a product of prime numbers only.
- Cubic centimetre -- the volume of a cube of edge one centimetre; 1,000 of them make one litre.
:::checkpoint Check yourself
- Work out 5³.
- Work out (−2)³.
- Find the cube root of 343 by factorisation.
- Between which two whole numbers does the cube root of 200 lie? :::
Bridge to practice
The exercises begin with direct cubing and recall of perfect cubes, then move to factorisation, sign reasoning and the scaling relationship between edge and volume. For any question involving a container, decide first whether you are moving from edge to volume or from volume to edge, since that single decision determines whether you cube or take the cube root.