💡 Cube Root vs Square Root
A square root asks: what number × itself = x? (2 times)
A cube root asks: what number × itself × itself = x? (3 times)
Also — cube roots can be negative! Square roots of negative numbers are not real.
The cube root of a number is the value that, when multiplied by itself three times, gives you that number.
If n × n × n = x, then ∛x = n
The small 3 in the symbol ∛ is called the index — it tells you how many times the number multiplies by itself.
2 factors
√9 = 3
3 factors
∛8 = 2
2³ = 2 × 2 × 2 = 8 ↔ ∛8 = 2
Cubing a number and taking its cube root cancel each other out.
If you can fill a cube with 8 small blocks arranged 2 × 2 × 2, then ∛8 = 2. The side length of the cube IS the cube root!
Cube roots can be negative! Because (−2) × (−2) × (−2) = −8. Three negatives multiply to a negative.
What number, multiplied by itself three times, equals 27? That number is ∛27.
A perfect cube is a number that has a whole-number cube root. 8, 27, 64, and 125 are all perfect cubes.
What is ∛216? (Think: what number × itself × itself = 216? Hint: try 6)
Each cube root is the side length of the 3D cube. The bigger the cube root, the bigger the cube shape. This is why it's called a "cube" root!
∛(a/b) = ∛a ÷ ∛b — find the cube root of the numerator, then find the cube root of the denominator. Works the same as square roots of fractions!
∛1 = 1 (because 1 × 1 × 1 = 1)
∛8 = 2 (because 2 × 2 × 2 = 8)
∛27 = 3 (because 3 × 3 × 3 = 27)
∛64 = 4 (because 4 × 4 × 4 = 64)
What is ∛(8/125)? (∛8 = ?, ∛125 = ?)
With square roots: √(−9) has no real answer (you cannot square a number and get a negative).
With cube roots: ∛(−8) = −2, because negative × negative × negative = negative.
Three negatives multiply to give a negative result!
∛(−8): the radicand is negative. This is okay for cube roots!
∛8 = 2 (because 2 × 2 × 2 = 8)
∛(−8) = −2. Check: (−2) × (−2) × (−2) = −8 ✅
🔑 Memory Rule
Negative under a cube root (∛) → the answer is negative. Just find the positive cube root first, then add the minus sign.
Negative under a square root (√) → no real answer on the GED.
What is ∛(−125)? (Find ∛125 first, then make it negative)
∛(xⁿ) — divide the exponent by 3.
If the exponent is divisible by 3: ∛(x³) = x, ∛(x⁶) = x²
The cube root and the ³ exponent cancel each other — just like division undoes multiplication.
The cube root and the exponent 3 cancel each other perfectly. ∛(x³) = x
If you cube x, you get x³. If you then take the cube root, you're undoing the cubing — you get back to x.
∛(x⁶) = x² (because 6 ÷ 3 = 2)
∛(x⁹) = x³ (because 9 ÷ 3 = 3)
What is ∛(64x³)? (∛64 = ?, ∛x³ = ?)