Square roots and cube roots help us undo squaring and cubing, two operations that appear often in geometry, measurement, algebra, and science. A square root tells the side length of a square when you know its area. A cube root tells the edge length of a cube when you know its volume.
Learning roots makes it easier to solve equations, estimate quantities, and understand formulas involving area and volume.
Roots are closely connected to powers and perfect numbers. For example, since 7^2 = 49, the square root of 49 is 7, and since 4^3 = 64, the cube root of 64 is 4. When a number is not a perfect square or perfect cube, you can estimate its root by comparing it to nearby perfect numbers.
Radical expressions can often be simplified by factoring out perfect square or perfect cube factors.
Key Facts
- Square root: if a^2 = b, then sqrt(b) = a for a nonnegative principal root.
- Cube root: if a^3 = b, then cbrt(b) = a.
- Perfect squares include 1, 4, 9, 16, 25, 36, 49, 64, 81, and 100.
- Perfect cubes include 1, 8, 27, 64, 125, 216, 343, 512, 729, and 1000.
- Simplifying square roots: sqrt(ab) = sqrt(a)sqrt(b), so sqrt(72) = sqrt(36 x 2) = 6sqrt(2).
- Estimating roots: since 64 < 70 < 81, sqrt(70) is between 8 and 9.
Vocabulary
- Square root
- A square root of a number is a value that, when multiplied by itself, gives the original number.
- Cube root
- A cube root of a number is a value that, when multiplied by itself three times, gives the original number.
- Perfect square
- A perfect square is a number that can be written as n^2 for an integer n.
- Perfect cube
- A perfect cube is a number that can be written as n^3 for an integer n.
- Radical expression
- A radical expression is an expression that contains a root symbol, such as sqrt(18) or cbrt(54).
Common Mistakes to Avoid
- Confusing sqrt(25) with 25/2 is wrong because a square root asks what number squared gives 25, so sqrt(25) = 5.
- Forgetting the negative solution in equations like x^2 = 49 is wrong because both 7^2 and (-7)^2 equal 49, so x = 7 or x = -7.
- Simplifying sqrt(50) as 25sqrt(2) is wrong because only the square root of the perfect square factor comes out, so sqrt(50) = sqrt(25 x 2) = 5sqrt(2).
- Treating cube roots like square roots for negative numbers is wrong because odd roots can be negative, so cbrt(-27) = -3.
Practice Questions
- 1 Simplify sqrt(98) completely.
- 2 Find cbrt(216) and explain how you know it is exact.
- 3 A square has area 45 square units and a cube has volume 45 cubic units. Explain why their side lengths are not found using the same type of root.