Radical Expressions

The "

" symbol is called the radical symbol. When a number is added under the radical symbol, for instance,

\sqrt{16}

, it becomes a radical expression.

\sqrt{16}

can be read as "the square root of 16", "root 16" or "radical 16".

Each radical expression has an index and a radicand.

\sqrt[(i n d e x)]{(r a d i c a n d)}

The square root is the most popular radical. For instance

\sqrt[2]{9}

has a radicand of 9 and index of 2. However, it is written as

\sqrt{9}

and the index of two is understood. "Squaring" a number and "taking the square root" are opposite operations.

3^{2} = 9 s o \sqrt{9} = 3

Numbers can be raised to powers other than two. For instance numbers can be cubed (raised to the 3^rd power), and raised to the 4^th, 5^th , 6^th and so on........ In the same way numbers can have different "roots".

Radical	Read as	Example	Opposite Operation
	Square root	$\sqrt{25} = 5$	5² = 25
	Cube root	$\sqrt[3]{8} = 2$	2³= 8
	Fourth root	$\sqrt[4]{81} = 3$	3⁴ = 81
	nth root	Used to indicate "any" root

Not all radical expressions have a "perfect" number, for instance

\sqrt{5}

. There is no number squared that will equal 5. Radical expressions can have an exact or an approximate answer. Exact answers will be left in radical form and approximate answers will be found with a calculator and will be a rounded decimal answer.

Radical expression	Simplified answer	Perfect/Non-Perfect
$\sqrt{36}$	6 because 6² = 36	Perfect square
$\sqrt[3]{27}$	3 because 3³ = 27	Perfect cube
$\sqrt{7}$	$\sqrt{7}$ answer in radical form 2.6 approximate answer	Non-Perfect

Radical expressions rely on a student having a strong multiplication background. Remember that "radical" and "root" mean the same thing. The index is the "root" and the number underneath is the radicand.

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Radical Expressions

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