Grade 10 — Mathematics 10C › C10U04 Factors, Roots & Irrational Numbers

Grade 10 › C10U04 Factors, Roots & Irrational Numbers › L3
GRADE 10 · ALBERTAOUTCOME AB-MAT10C-AN1

L3 · Perfect squares, perfect cubes, and their roots

Learning goal

“I can decide — concretely, from a prime factorization — whether a whole number is a perfect square, a perfect cube, both, or neither, and I can find the exact square root of a perfect square and the exact cube root of a perfect cube.”

Perfect square — a whole number that is the square of a whole number: 36 = 6². Pictured as a square array of dots.
Perfect cube — a whole number that is the cube of a whole number: 27 = 3³. Pictured as a cube built from unit cubes.
Square root — √n asks: "what non-negative number, squared, gives n?" √36 = 6. The symbol √ always means this principal (non-negative) root.
Cube root — ∛n asks: "what number, cubed, gives n?" ∛27 = 3, and ∛(−27) = −3: cube roots of negatives exist, because a negative cubed is negative.
Worked example

Is 216 a perfect square? A perfect cube? Step 1: Factor: 216 = 2³ × 3³. Step 2: Square test — in a perfect square, EVERY exponent is even (the number is built from pairs). 216's exponents are 3 and 3, both odd -> NOT a perfect square. Step 3: Cube test — in a perfect cube, EVERY exponent is a multiple of 3 (the number is built from triples). 3 and 3 qualify -> 216 IS a perfect cube. Step 4: Its cube root takes one from each triple: ∛216 = 2 × 3 = 6. Check: 6³ = 216 ✓ And 144 = 2⁴ × 3²: exponents 4 and 2 are both even -> perfect square, √144 = 2² × 3 = 12 (halve the exponents). But 2 is not a multiple of 3 -> not a perfect cube. Say it plainly: pairs for squares, triples for cubes — read the exponents.

Check yourself — explain it

Why does halving the exponents give the square root of a perfect square?

Paired problem — example still visible

Decide for 729 = 3⁶: perfect square? perfect cube? Give every exact root. -> Both. Exponent 6 is even -> perfect square, √729 = 3³ = 27 (check: 27² = 729 ✓). Exponent 6 is a multiple of 3 -> perfect cube, ∛729 = 3² = 9 (check: 9³ = 729 ✓).

Faded completion — last step is yours

"512 = 2⁹. The exponent 9 is a multiple of ___, so 512 is a perfect ___; ∛512 = 2^__ = ___. It is not a perfect square because 9 is not ___." -> 3; cube; 3; 8; even

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