Cubes and Cube Roots
Numbers · Cubes and Cube Roots
Syllabus tag: KCSE | Mathematics | Form 2 | Topic 1 Cubes and Cube Roots
Lesson objectives
By the end of this topic, you should be able to:
- Find the cube of a number by multiplication.
- Find the cube root of a number by the factor method.
- Find cubes and cube roots from mathematical tables.
- Apply cubes and cube roots to real situations.
Cubes and Cube Roots
Cubing multiplies a number by itself three times. The cube root reverses that.
a) Cubing
Write it with an index 3. So 4³ means 4 × 4 × 4, which is 64.
Note that 4³ is 64, not 12. Cubing is not tripling.
The name comes from volume: a cube of side 4 units has a volume of 64 cubic units.
b) Cubes of negatives, fractions and decimals
A negative number cubed stays negative, because three negative factors give a negative result. So (−3)³ = −27.
This differs from squaring, where a negative always becomes positive. That difference matters when solving equations.
For a fraction, cube the numerator and the denominator. So (2/3)³ = 8/27.
For a decimal, multiply out and count the places. So 0.2³ = 0.008, with three decimal places.
c) Perfect cubes
The first ten are 1, 8, 27, 64, 125, 216, 343, 512, 729 and 1000.
Knowing these on sight makes cube roots by inspection much quicker.
d) Cube root by the factor method
Write the number as a product of primes in index form.
Divide every index by 3. The result is the cube root.
This works because cubing multiplies each index by 3, so rooting must divide by 3.
If any index is not divisible by 3, the number is not a perfect cube.
e) Cube root of a negative
Every real number has exactly one real cube root, including negatives.
So ∛−27 = −3, because (−3)³ = −27.
This differs from square roots, where a negative number has no real square root at all.
f) Using tables
Mathematical tables give cubes and cube roots to four figures.
Read the row for the whole part and the column for the first decimal. Then apply the difference correction.
Estimate first. Since 10³ = 1000 and 11³ = 1331, ∛1200 must lie between 10 and 11.
g) Where this is used
Volume problems, where a cube's side comes from its volume. Scaling solids, where volume changes with the cube of the linear factor. Formulas in physics and engineering.
Words to know
- Cube -- the result of multiplying a number by itself three times.
- Perfect cube -- a number whose cube root is a whole number.
- Cube root -- a value which, when cubed, gives the original number.
- Index -- the raised number showing how many times a base is multiplied by itself.
- Mean differences -- the correction columns in mathematical tables.
:::checkpoint Check yourself
- Work out 6³.
- Work out (−2)³.
- Find ∛3375 by the factor method.
- Why does dividing the indices by 3 give the cube root? :::
Bridge to practice
The exercises begin with cubes and the cubes of negatives, move through perfect cubes and the factor method, and finish with tables, estimation and decimals. Before accepting any cube root answer, name the two perfect cubes it lies between.