Squares & Square Roots explained through a friendly whiteboard conversation, examples, visuals, and practice for school students.
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Notes
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Formulae
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Examples
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A square is the result of multiplying a number by itself.
A square root works in the opposite direction. It asks which number was multiplied by itself.
Squares and square roots are used in area, geometry, Pythagoras theorem, estimation, and algebra.
The square of 7 is 7 x . We write this as .
A perfect square is a number that is the square of an integer. Examples are 1, 4, 9, 16, 25, and 36.
The square root of 49 is 7 because 7 x .
Prime factorisation helps find square roots because factors in a perfect square can be paired.
If one prime factor is unpaired, the number is not a perfect square.
Meaning: A square is a number multiplied by itself.
Definition: The square of a is x a.
x .
Meaning: A perfect square has an integer square root.
Definition: A perfect square is the product of an integer with itself.
81 is a perfect square because 9 x .
Meaning: Square root is the number that gives the original number when squared.
Definition: The square root of n is a number whose square is n.
The square root of 64 is 8 because .
Meaning: This method uses prime factors to find square roots.
Definition: Prime factorisation writes a number as a product of prime factors.
x 2 x 2 x 2 x 3 x 3. Pair the factors: (2 x 2) and (2 x 2) and (3 x 3). Square roo x 2 x .
Meaning: The smallest multiplier can make all prime factors pair up.
Definition: It is the least number multiplied with a given number to make it a perfect square.
If x 2 x 2 x 2 x 3, the factor 3 is unpaired. Multiply by 3 to get a perfect square.
Easy
Question: Find .
Medium
Question: Find the square root of 196 by prime factorisation.
Exam-level
Question: Find the smallest number by which 72 must be multiplied to make a perfect square.
Thinking is 18
Why it happens: Squaring is confused with multiplying by 2.
Correct approach: means 9 x .
Not pairing all prime factors
Why it happens: The factorisation is not checked.
Correct approach: Pair equal prime factors before taking the root.
Using an unpaired factor in the square root
Why it happens: All factors are multiplied directly.
Correct approach: Take one factor from each complete pair only.
Is every number a perfect square?
No. Only numbers that are squares of integers are perfect squares.
Why do prime factors need pairs?
A square is made by multiplying a number by itself, so its prime factors repeat in pairs.
What is the square root of 1?
The square root of 1 is 1 because 1 x .
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