Simplifying Radicals
Simplifying Radicals
A radical (or root) is the inverse of an exponent. The square root of 25 is 5 because 5² = 25. Simplifying radicals means writing them in their simplest form by extracting perfect-square factors from under the radical sign.
For example, the square root of 72 simplifies to 6 times the square root of 2 because 72 equals 36 times 2, and the square root of 36 is 6. This process is called simplifying or "reducing" the radical.
This lesson teaches you to simplify square roots, cube roots, and higher-order radicals, building the foundation for radical operations and equations.
A radical expression involves a root — most commonly a square root, but also cube roots, fourth roots, and so on.
The principal $n$th root of $a$ is written $\sqrt[n]{a}$. For square roots we write $\sqrt{a}$ (the index 2 is understood). The number under the radical sign is called the radicand.
If $a \ge 0$ and $b \ge 0$, then: $$\sqrt{ab} = \sqrt{a} \cdot \sqrt{b}$$
A square root is in simplest form when the radicand has no perfect-square factor other than 1. To simplify:
- Factor the radicand into a product that includes the largest perfect square.
- Apply the product rule: $\sqrt{a \cdot b} = \sqrt{a}\cdot\sqrt{b}$.
- Simplify the perfect-square root.
Simplify $\sqrt{72}$.
Find the largest perfect square that divides 72: $72 = 36 \cdot 2$.
$$\sqrt{72} = \sqrt{36 \cdot 2} = \sqrt{36}\cdot\sqrt{2} = 6\sqrt{2}$$
Simplify $\sqrt{200}$.
$200 = 100 \cdot 2$.
$$\sqrt{200} = \sqrt{100}\cdot\sqrt{2} = 10\sqrt{2}$$
Simplify $\sqrt{48}$.
$48 = 16 \cdot 3$.
$$\sqrt{48} = \sqrt{16}\cdot\sqrt{3} = 4\sqrt{3}$$
Simplify $\sqrt{18x^4}$ where $x \ge 0$.
$18x^4 = 9x^4 \cdot 2$.
$$\sqrt{18x^4} = \sqrt{9x^4}\cdot\sqrt{2} = 3x^2\sqrt{2}$$
Simplify $\sqrt{75a^3b^2}$ where $a, b \ge 0$.
$75a^3b^2 = 25a^2b^2 \cdot 3a$.
$$\sqrt{75a^3b^2} = 5ab\sqrt{3a}$$
Cube Roots and Higher-Index Radicals
$$\sqrt[3]{a \cdot b} = \sqrt[3]{a} \cdot \sqrt[3]{b}$$
Factor out perfect cubes to simplify.
Simplify $\sqrt[3]{54}$.
$54 = 27 \cdot 2$.
$$\sqrt[3]{54} = \sqrt[3]{27}\cdot\sqrt[3]{2} = 3\sqrt[3]{2}$$
Simplify $\sqrt[3]{-128}$.
$-128 = -64 \cdot 2$.
$$\sqrt[3]{-128} = \sqrt[3]{-64}\cdot\sqrt[3]{2} = -4\sqrt[3]{2}$$
Practice Problems
Simplify each radical. Assume all variables are non-negative.
Show Answer Key
1. $5\sqrt{2}$
2. $7\sqrt{2}$
3. $8\sqrt{2}$
4. $6\sqrt{5}$
5. $2x\sqrt{3}$
6. $3a^2\sqrt{5b}$
7. $11\sqrt{2}$
8. $10\sqrt{5}$
9. $2\sqrt[3]{5}$
10. $-5\sqrt[3]{2}$
11. $4x^3y\sqrt{2y}$
12. $12\sqrt{2}$
13. $6$
14. $5ab^2\sqrt{6}$
15. $3\sqrt[4]{2}$