RIN - Simplify Radical Expressions (Lesson)

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Simplify Radical Expressions

Square Roots

If a squared equals b, then a is the square root of b.  

Let's think of some common perfect squares:

x 1 2 3 4 5 6 7 8 9 10 11 12
x2 1 4 9 16 25 36 49 64 81 100 121 144

 

 

the square root symbol is called a radical and the number or element under the square root is called the radicand

Examples:

  1. LaTeX: \sqrt[2]{25}=52√25=5

  2. LaTeX: \sqrt[2]{121}=112√121=11

  3. LaTeX: \sqrt[2]{-16}=\:does\:not\:exist2√−16=doesnotexist

LaTeX: \sqrt[2]{-16}2√−16   does not have an answer because 42 = 16 and (-4)2 = 16, there is no number that you can multiply by itself to get -16.

Product Property is square root of a times b equals the square root of a times the square root of b. The quotient property is the square root of a divided by b which equals the square root of a divided by the square root of b.

Simplifying Square Root Examples:

  1. LaTeX: \sqrt[2]{\frac{64}{100}}=\frac{\sqrt[2]{64}}{\sqrt[2]{100}}=\frac{8}{10}=\frac{4}{5}2√64100=2√642√100=810=45

  2. LaTeX: \sqrt[2]{18}\cdot\sqrt[2]{2}=\sqrt[2]{36}=62√18⋅2√2=2√36=6

  3. LaTeX: \sqrt[2]{\frac{100}{25}}=\sqrt[2]{4}=22√10025=2√4=2

Let's look at some examples that are not perfect squares:

Cube Roots

A cube root is similar to a square root, but instead of having two roots that multiply to the radicand, there are THREE roots. For example, 2 • 2 • 2 = 8, so LaTeX: \sqrt[3]{8}=23√8=2.

Let's think of some common perfect cubes:

x 1 2 3 4 5
x3 1 8 27 64 125

 

 

Examples:

  1. LaTeX: \sqrt[3]{125}=53√125=5

  2. LaTeX: \sqrt[3]{27}=33√27=3

  3. LaTeX: \sqrt[3]{-64}=\:-43√−64=−4

LaTeX: \sqrt[3]{-64}3√−64   DOES exist because -4 • -4 • -4 = -64. Cube roots CAN be negative. 

Simplifying Cube Root Examples:

  1. LaTeX: \sqrt[3]{\frac{8}{125}}=\frac{\sqrt[3]{8}}{\sqrt[3]{125}}=\frac{2}{5}3√8125=3√83√125=25

  2. LaTeX: \sqrt[3]{4}\cdot\sqrt[3]{16}=\sqrt[3]{64}=43√4⋅3√16=3√64=4

  3. LaTeX: \sqrt[3]{40}=\sqrt[3]{8}\cdot\sqrt[3]{5}=2\sqrt[3]{5}3√40=3√8⋅3√5=23√5

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