Reducing Radicals and Square Roots

Radicals and Square Roots

Introduction to Radicals

A radical is a mathematical symbol used to indicate the root of a number. The expression under the radical sign is called the radicand. If there is a number as a superscript outside the radical, it is called the index. If no index is shown, the radical is assumed to be a square root (index 2).

• Radical Symbol: $\sqrt{\ }$

• Radicand: The value inside the radical symbol.

• Index: The small number outside and to the upper left of the radical symbol (e.g., $\sqrt[3]{8}$ has index 3).

• Square Root: If no index is written, the index is 2 by default.

Example: In $\sqrt{16}$, 16 is the radicand, and the index is 2 (implied).

Definition of Square Root

The square root of a number is a value that, when multiplied by itself, gives the original number.

• For example, $2 \times 2 = 4$, so $\sqrt{4} = 2$.

• Similarly, $3 \times 3 = 9$, so $\sqrt{9} = 3$.

Perfect Squares

A perfect square is a number that is the square of an integer. Knowing perfect squares helps in simplifying radicals.

Integer	Perfect Square
2	4
3	9
4	16
5	25
6	36
7	49
8	64
9	81
10	100
11	121
12	144

List of perfect squares: 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, ...

Finding Perfect Square Roots

To find the square root of a perfect square, determine which integer squared gives the radicand.

• Example: $\sqrt{25} = 5$ because $5 \times 5 = 25$.

• Example: $\sqrt{81} = 9$ because $9 \times 9 = 81$.

Reducing Radicals

Steps to Reducing a Radical

When the radicand is not a perfect square, we can simplify the radical by factoring out perfect squares.

1. Use the list of perfect squares to find the largest perfect square that divides evenly into the radicand.

2. Rewrite the radicand as a product of the perfect square and another factor.

3. Take the square root of the perfect square and move it outside the radical. The remaining factor stays under the radical.

Formula:

$\sqrt{ab} = \sqrt{a} \times \sqrt{b}$

where a is the largest perfect square factor of the radicand.

Example of Reducing a Radical

• Example: Reduce $\sqrt{72}$.

• Step 1: The largest perfect square that divides 72 is 36 ($36 \times 2 = 72$).

• Step 2: Rewrite as $\sqrt{36 \times 2}$.

• Step 3: $\sqrt{36 \times 2} = \sqrt{36} \times \sqrt{2} = 6\sqrt{2}$.

Practice Examples

• Example 1: $\sqrt{50} = \sqrt{25 \times 2} = 5\sqrt{2}$

• Example 2: $\sqrt{18} = \sqrt{9 \times 2} = 3\sqrt{2}$

• Example 3: $\sqrt{12} = \sqrt{4 \times 3} = 2\sqrt{3}$

• Example 4: $\sqrt{27} = \sqrt{9 \times 3} = 3\sqrt{3}$

Summary Table: Reducing Common Radicals

Radical	Factorization	Reduced Form
$\sqrt{8}$	$\sqrt{4 \times 2}$	$2\sqrt{2}$
$\sqrt{20}$	$\sqrt{4 \times 5}$	$2\sqrt{5}$
$\sqrt{32}$	$\sqrt{16 \times 2}$	$4\sqrt{2}$
$\sqrt{45}$	$\sqrt{9 \times 5}$	$3\sqrt{5}$
$\sqrt{72}$	$\sqrt{36 \times 2}$	$6\sqrt{2}$

Key Points to Remember

• Always look for the largest perfect square factor of the radicand.

• Only the square root of perfect squares can be taken out of the radical.

• Leave any remaining non-perfect square factor under the radical.

Additional info: In higher mathematics, these techniques extend to cube roots, fourth roots, and beyond, but the process of factoring out perfect powers remains similar.