Simplifying Radical Expressions

How to simplify a radical

Goal: Extract all perfect square factors from under the radical. $\sqrt{a^2 \cdot b} = a\sqrt{b}$.

Step-by-step: Simplify $\sqrt{72x^5y^3}$

Step 1 — Factor the number into prime factors and find perfect squares:

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

Step 2 — Factor the variables by extracting even powers:

$x^5 = x^4 \cdot x = (x^2)^2 \cdot x$

$y^3 = y^2 \cdot y = (y)^2 \cdot y$

Step 3 — Rewrite under one radical:

$\sqrt{72x^5y^3} = \sqrt{6^2 \cdot (x^2)^2 \cdot y^2 \cdot 2xy}$

Step 4 — Extract perfect squares (each pair comes out as a single factor):

$= 6 \cdot x^2 \cdot y \cdot \sqrt{2xy} = 6x^2y\sqrt{2xy}$

Final answer: $6x^2y\sqrt{2xy}$

The rule for variables

For $\sqrt{x^n}$: divide $n$ by 2. The quotient goes outside, the remainder stays inside.

$\sqrt{x^5}$: $5 \div 2 = 2$ remainder $1$. So $\sqrt{x^5} = x^2\sqrt{x}$.

Common mistakes

• Forgetting to simplify the number. $\sqrt{72} \neq \sqrt{72}$. Always check: $72 = 4 \times 18 = 4 \times 9 \times 2$. Biggest perfect square factor: $36$.
• Wrong exponent extraction. $\sqrt{x^5}$ extracts $x^2$ (not $x^{2.5}$).

Try it in the visualization

Enter any number. The prime factorization tree builds, factors pair up, and perfect squares are extracted step by step.