L2 · Perfect Squares and Square Roots (within 144)

Learning goal

“I can list the perfect squares to 144, write any square number as repeated multiplication and as a power, and explain why a positive number has two square roots, zero has one, and a negative number has none (in the real numbers).”

Perfect square — a number that is a whole number times itself: 36 = 6 x 6. The perfect squares within 144 are 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144.

Square root — the number that, multiplied by itself, gives the number under the sign: the square root of 36 is 6 because 6 x 6 = 36.

Principal square root — the positive square root, the one the √ sign names: √36 = 6 (never -6). The negative root is written -√36 = -6.

Radicand — the number under the radical sign: in √36, the radicand is 36.

Worked example

Is 72 a perfect square? Step 1: Run the family list: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144. Step 2: 72 is not on the list. 8^2 = 64 < 72 < 81 = 9^2, and there is no whole number between 8 and 9 - so no whole number multiplies by itself to give 72. Step 3: 72 is a non-perfect square. And a non-perfect square still has an exact "repeated multiplication" form: 72 = √72 x √72 = (√72)^2 - the product of two identical IRRATIONAL numbers (this is the curriculum's definition of a non-perfect square). Say it plainly: every positive number n can be written as a power two ways in this unit - perfect squares as whole-number powers (64 = 8^2), non-perfect squares as radical powers (72 = (√72)^2).

Check yourself — explain it

Why is 49 a perfect square but 50 not?

Paired problem — example still visible

Classify 121, and write it as repeated multiplication and as a power. -> Perfect square: 121 = 11 x 11 = 11^2.

Faded completion — last step is yours

"144 = ___ x ___ = ___^2, so 144 is a ___________ square." -> 12; 12; 12; perfect

Ready to practise?

10 scored questions with a 3-rung hint ladder. Mastery needs 8/10 plus both gate questions.

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