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.
More to read
1.8 The Real Numbers
Numbers like 4, 9, and 16 are perfect squares because you get them by multiplying a whole number by itself: 2×2=4, 3×3=9, 4×4=16. Finding a square root means going backwards: what number, multiplied by itself, gives you this number? So √16 = 4, because 4×4=16.
Not every number has a whole-number square root. √16 is exact, but √17 is not. We can only give a decimal approximation for it, like √17 ≈ 4.123. That decimal never settles into a neat repeating pattern, so it's called irrational.
Every number you can think of, whether a whole number, a fraction, or an endless decimal, fits somewhere on the number line.
Key ideas: unit fraction exponents
You already know that squaring a number means multiplying it by itself, like 5²=25. Roots work the opposite way: the square root of 25 asks, what number times itself gives 25? The answer is 5.
Cube roots work the same way, but for three copies of a number multiplied together. Since 2×2×2=8, the cube root of 8 is 2.
Mathematicians also write roots using fraction exponents. b to the power of one half means the square root of b, and b to the power of one third means the cube root of b. This keeps all the exponent rules working the same way, whether the exponent is a whole number or a fraction. So instead of learning a totally separate set of rules for roots, you can just think of them as another kind of power.
Higher order roots
A square root undoes squaring, and a cube root undoes cubing. Since 3²=9, the square root of 9 is 3. Since 3³=27, the cube root of 27 is 3.
Higher order roots follow the same idea. A fourth root undoes raising a number to the fourth power, a fifth root undoes the fifth power, and so on. If 2⁴=16, then the fourth root of 16 is 2.
Perfect squares like 1, 4, 9, 16, and 25, and perfect cubes like 1, 8, 27, and 64, give exact whole-number roots. Most other numbers don't, so their roots are irrational and can only be written as decimal approximations that never settle into a repeating pattern.
Numbers and operations: FAQ
Square roots and cube roots show up everywhere in math. If you want the side length of a square with a certain area, you take the square root of that area. If you want the side length of a cube with a certain volume, you take the cube root of that volume.
Some numbers give exact whole-number answers: the square root of 36 is exactly 6, because 6×6=36. Other numbers, like 40, don't have a whole-number square root, so we can only write a decimal that gets close to the real answer, like √40 ≈ 6.32.
Explore more
2 · Watch
به انگلیسیTake it off screen
Where it sits
Learn first
Nothing; this is a starting point.
Where this leads
Jobs that lean on this skill. Follow one to see everything it is built on.
14 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.