Simplifying With Prime Factors
Simplify radical expressions in Grade 9 algebra using prime factorization: break radicand into prime factors, group pairs under the radical, and extract perfect square factors outside.
Key Concepts
Property Any non negative integer can be expressed as a product of prime numbers. Use this factorization under the radical, then apply the property $\sqrt{ab} = \sqrt{a} \cdot \sqrt{b}$. Explanation Can't spot a perfect square? No problem! Break the number down to its prime building blocks. Look for pairs of identical prime numbers. For every pair you find (like two 5s), one gets to escape the radical sign. Any prime without a partner has to stay behind. It’s like Noah’s Ark for numbers! Examples $\sqrt{98} = \sqrt{2 \cdot 7 \cdot 7} = \sqrt{2} \cdot \sqrt{7^2} = 7\sqrt{2}$ $\sqrt{150} = \sqrt{2 \cdot 3 \cdot 5 \cdot 5} = \sqrt{2 \cdot 3} \cdot \sqrt{5^2} = 5\sqrt{6}$.
Common Questions
How do you simplify a radical using prime factorization?
Find the prime factorization of the radicand. Under a square root, pair identical prime factors. Each pair comes out of the radical as one factor. Any unpaired primes stay under the radical.
How do you simplify √72 using prime factors?
Prime factorize 72: 2 × 2 × 2 × 3 × 3. Under √: pair the 2s: (2×2) and (3×3) with one 2 remaining. Pull out pairs: 2 × 3 = 6 outside, √2 inside. Result: 6√2.
Why is prime factorization more reliable than looking for perfect square factors?
Prime factorization is systematic and works every time, even for large numbers. Searching for perfect square factors requires pattern recognition that can miss factors. Prime factorization guarantees the fully simplified form.