Simplifying radicals involves rewriting a square root (or other radical) expression in its simplest form. This often includes removing factors of the radicand that are perfect squares and ensuring there are no radicals in the denominator of a fraction. Here's how you simplify radicals step by step:

Steps to Simplify Radicals:

1. Find the prime factorization of the number inside the radical. Break the radicand (the number inside the radical) into its prime factors.

2. Pair factors for square roots. For a square root, if a factor appears in pairs, one instance of that factor can come out of the radical.

3. Multiply factors outside the radical. Combine any factors that have been removed from the radical.

4. Simplify any remaining radical. Ensure the remaining number inside the radical has no square factors.

5. Rationalize the denominator (if the radical is part of a fraction). Multiply the numerator and denominator by the radical in the denominator to eliminate it.

Example 1:

Simplify $\sqrt{72}$.

1. Prime factorization of 72: $72 = 2 \cdot 2 \cdot 2 \cdot 3 \cdot 3$

2. Pair the factors: $(2 \cdot 2) \text{ and } (3 \cdot 3)$

3. Take one from each pair out of the radical: $2 \cdot 3 = 6$

4. Simplify: $\sqrt{72} = 6\sqrt{2}$

Example 2:

Simplify $\frac{5}{\sqrt{3}}$.

1. Rationalize the denominator: Multiply numerator and denominator by $\sqrt{3}$: $\frac{5}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} = \frac{5\sqrt{3}}{3}$

2. Simplified form: $\frac{5}{\sqrt{3}} = \frac{5\sqrt{3}}{3}$

Let me know if you'd like more examples or further clarification on any step!

Relative Questions:

1. How do you simplify cube roots or higher roots?
2. What is the difference between rationalizing the denominator for square roots and higher roots?
3. How do you handle simplifying radicals with variables, like $\sqrt{x^3}$?
4. Can radicals be added or subtracted, and how does simplification help with this?
5. How do you simplify radicals involving fractions, like $\sqrt{\frac{49}{16}}$?

Tip:

Always check if the radicand can be factored further after an initial simplification to ensure you have the simplest form!