How do you simplify radical expressions?

Simplifying Radical Expressions

Definition:
To simplify a radical expression means to rewrite it so that:

• The radicand (the number inside the radical) has no perfect square factors (other than 1) if it's a square root.
• There are no radicals in the denominator (if possible).
• The expression is as simple as possible.

The most common radical is the square root, but the same principles apply to cube roots and higher.

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Example: Simplify $\sqrt{72}$

Step 1: Factor the radicand into primes.

$$ 72 = 2 \times 36 = 2 \times 6 \times 6 = 2 \times 2 \times 3 \times 2 \times 3 = 2^3 \times 3^2 $$

Step 2: Identify perfect squares.

• $2^2 = 4$ is a perfect square. - $3^2 = 9$ is a perfect square. **Step 3: Rewrite the radical using these factors.** %%MATH_DISPLAY_1%% **Step 4: Separate perfect squares from the rest.** %%MATH_DISPLAY_2%% **Step 5: Simplify each square root.** - $sqrt{2^2} = 2$ - $sqrt{3^2} = 3$ - $sqrt{2}$ remains

So,

$$ \sqrt{72} = 2 \times 3 \times \sqrt{2} = 6\sqrt{2} $$

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Takeaways

• Factor the radicand to find perfect powers matching the root.
• Move perfect powers outside the radical.
• The simplified form has no perfect square factors (for square roots) inside the radical.