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Simplify radicals containing perfect-square factors

A learner understands that a radical can be simplified by factoring its nonnegative radicand into a perfect-square factor and a remaining factor, extracting the square root of the perfect square: 72=36⋅2=62\sqrt{72}=\sqrt{36\cdot2}=6\sqrt2. The learner preserves equivalent value, recognizes that sums cannot be split in the same way, and writes numerical radicals in simplest form; this scope excludes more advanced symbolic, higher-index, or complex-radical simplification.

Detailed Explanation: Simplify radicals containing perfect-square factors

A radical is in simplest form when its radicand has no perfect-square factor greater than 11.

To simplify a numerical radical:

  1. Find a perfect-square factor of the radicand.
  2. Rewrite the radicand as a product.
  3. Use ab=ab\sqrt{ab}=\sqrt a\sqrt b.
  4. Take the square root of the perfect square.

Example: Simplify 72\sqrt{72}.

First, factor (72)(72) using the largest perfect square factor:

72=36⋅272=36\cdot 2

Rewrite the radical:

72=36⋅2\sqrt{72}=\sqrt{36\cdot 2}

Separate the product under the radical:

36⋅2=362\sqrt{36\cdot 2}=\sqrt{36}\sqrt{2}

Since 36=6\sqrt{36}=6, simplify:

72=62\boxed{\sqrt{72}=6\sqrt{2}}

The answer is in simplest form because 22 has no perfect-square factor greater than 11.

Remember, this property applies to multiplication inside a radical, not addition. For example, 36+2\sqrt{36+2} cannot be changed to 36+2\sqrt{36}+\sqrt{2}.

Learn by doing: Simplify radicals containing perfect-square factors

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    Radicals - Square - Simplifying, Values only, Radical Remaining

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