Integer powers and roots
Use integer powers and associated real roots (square, cube, and higher); recognise powers of 2, 3, 4, and 5; distinguish between exact representations of roots and their decimal approximations
What a learner can do afterwards
- Calculate squares, cubes, and higher integer powers of whole numbers
- Find square roots and cube roots of perfect squares and perfect cubes
- Recognise key powers (powers of 2 up to 2¹⁰, powers of 3 up to 3⁵, etc.) and distinguish exact roots from approximations
1 · Read
A square number comes from multiplying a whole number by itself. We write this as n², which means n × n. For example, 4² = 4 × 4 = 16.
A cube number comes from multiplying a whole number by itself three times. We write this as n³, which means n × n × n. For example, 10³ = 10 × 10 × 10 = 1000.
Priya wants to find 5² and 3³. She works out 5² = 5 × 5 = 25, and 3³ = 3 × 3 × 3 = 27. Squaring and cubing start the same way, but cubing multiplies by the number one more time.
The small raised number is the power. It counts how many copies of the number you multiply, so 2⁴ = 2 × 2 × 2 × 2 = 16. Learn the key ones: the powers of 2 double each time, 2, 4, 8, 16, 32, 64, and the powers of 3 are 3, 9, 27, 81, 243.
Going backwards from a power is called finding a root. The square root symbol √ asks: what number times itself gives this? √25 = 5, because 5² = 25. The cube root symbol ∛ asks: what number multiplied by itself three times gives this? ∛27 = 3, because 3³ = 27.
Not every root is exact. √25 = 5 exactly, because 5² = 25. But √50 falls between 7 and 8, because 7² = 49 and 8² = 64, so we say it is close to 7 rather than exact.
Squaring multiplies a number by itself once, cubing multiplies it twice more, and the root symbols √ and ∛ undo them, though not every root comes out as a whole number.
2 · Watch
Take it off screen
Where it sits
Where this leads
Jobs that lean on this skill. Follow one to see everything it is built on.
8 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.