Use prime factorization to find √196
- 196 = 2 × 98 = 2 × 2 × 49 = 2² × 7².
- Take one factor from each equal pair: 2 × 7 = 14.
- Check: 14² = 196.
Factors & prime factorization
Find the equal factors hidden inside a square.
Factor pairs describe rectangular arrays
Use factor pairs and prime-factor pairs to recognize perfect squares and find their roots.
A positive factor divides a whole number exactly. The factor pairs of 36 are 1 × 36, 2 × 18, 3 × 12, 4 × 9 and 6 × 6. The equal middle pair gives the square root. A positive perfect square has an odd number of positive factors because its equal pair contributes only one distinct middle factor.
Explore the relationship
Predict first. Change one value, then explain the result. Diagrams show the entered dimensions; displayed decimals are rounded.
Prime factors break a number into basic products
A prime number is a whole number greater than 1 with exactly two positive factors: 1 and itself. The number 1 is not prime. Prime factorization writes a whole number greater than 1 as a product of primes. For example, 72 = 2 × 2 × 2 × 3 × 3 = 2³ × 3².
Pair prime factors to recover a square root
For a perfect square, every prime factor occurs an even number of times. Arrange one from each pair along each side. Since 144 = 2⁴ × 3², its square root is 2² × 3 = 12. An unpaired factor shows that a positive integer is not a perfect square. The perfect square 1 is a special case: its square root is 1 and it has no prime factors.
Worked examples
Is 180 a perfect square?
- 180 = 2² × 3² × 5.
- The factor 5 is unpaired, so 180 is not a perfect square.
- Since 13² = 169 and 14² = 196, its root lies between 13 and 14.
Guided Practice
Find √400 using prime factors.
A first hint
400 = 4 × 100 = 2⁴ × 5².
Compare your reasoning
Take one from each pair: 2² × 5 = 20. Check 20² = 400.
Practice
Fresh questions · useful feedback
Build fluency, one decision at a time.
Work through eight questions. Two independent correct answers introduce a harder version; supported answers bring the next question back to the foundation. You can use hints at any time.
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Connections
List the positive factors of 36 and 40. Explain the odd/even difference in their factor counts without relying only on the lists.
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