How to Simplify Radicals
Simplifying radicals means rewriting a radical expression in its simplest form, where the radicand (the number inside the radical) has no perfect square factors (for square roots) or no perfect cube factors (for cube roots), etc., other than 1.
Steps to Simplify Square Roots
- Find the prime factorization of the number inside the square root.
- Identify pairs of factors (for square roots, pairs of the same number).
- Take one number from each pair outside the radical.
- Multiply the numbers outside the radical.
- Multiply the leftover factors inside the radical.
- Write the simplified expression.
Example
Simplify 72\sqrt{72}72:
- Prime factorization of 72: 72=2×2×2×3×372=2\times 2\times 2\times 3\times 372=2×2×2×3×3
- Group pairs: (2×2)(2\times 2)(2×2) and (3×3)(3\times 3)(3×3)
- Take one number from each pair outside: 222 and 333
- Multiply outside: 2×3=62\times 3=62×3=6
- Leftover inside: one 222 remains inside
- Simplified form: 626\sqrt{2}62
Tips for Other Roots
- For cube roots, look for triplets of the same number.
- For fourth roots, look for groups of four identical factors, and so on.
If you want, I can provide examples for cube roots or higher roots too!
