Radicals and Complex Numbers: Simplify Radical Expressions (Lesson 2.03)

Lesson Goals: 

    0. Simplify radical expressions whose radicands contain perfect square factors.

  1. Simplify radical expressions by rationalizing the denominator.

  2. Multiply radical expressions.

  3. Add and subtract radical expressions.

  4. Simplify radical expressions.

What is a radical expression? To put it in simple terms, it’s an expression with a radical sign like this (btw it doesn’t need the 4) 4\sqrt4

What is the radicand? A radicand is the number, value, or expression under the radical sign. Examples: 5−∪−857\sqrt5-\sqrt{\cup-857}  The radicands in these two examples are 5 and u-875

What is the Index? It’s the small number written to the left of the radical sign, which is also called the root. If there is no written index, it’s 2.  Example:  96\sqrt[6]{9}  The index is 6

When reading radical expressions, you would use the square root of blank for 2, the cube root of blank for 3, and __th root of blank for any other number. 

Rules for simplifying a radical expression.  1. The radicand cannot be a fraction. 2. No radicals can be in the denominator. 3. No factor of the radicand can be a perfect square other than 1. To be in simple Radical form, it must follow all three of these rules.

Steps to simplify a radical expression

1.) Find all perfect square factors under the radical. To do that, use the property of ab=a⋅b\sqrt{ab}=\sqrt{a}\cdot\sqrt{b}

 Follow along problem 72\sqrt{72}

72 has many factors, but we are only interested in the ones that are a perfect square. Depending on which one you use, they can be solved differently. Choosing a pair of factors that have the largest perfect square can use fewer steps.

For this equation, if you use the largest perfect square factor, 36, we will only need to simplify once

72=36⋅2\sqrt{72}=\sqrt{36}\cdot\sqrt2

=6⋅2=6\cdot\sqrt2

=62=6\sqrt2

For this equation, if you use a smaller perfect square factor,9, the process will need to be repeated until the radicand is completely simplified.

72=9⋅8\sqrt{72}=\sqrt9\cdot\sqrt8

=3⋅8=3\cdot\sqrt8 This is not completely simplified because it has a perfect square factor of 4

=3.4⋅2=3.\sqrt4\cdot\sqrt2  

=3⋅2⋅2=3\cdot2\cdot\sqrt2

=62=6\sqrt2

Simplifying a fractional radicand can be done using the property ab=ab\sqrt{\frac{a}{b}}=\frac{\sqrt{a}}{\sqrt{b}}

Example: Simplify the radicand 259\sqrt{\frac{25}{9}}

=259\frac{\sqrt{25}}{\sqrt9}

53\frac53

Rationalizing the denomaniaotr

Some expressions will need to be rewritten so that there is no radical in the denominator. The process is called rationalizing the denominator. Before rationalizing the denominator, you should ask these questions: 1.) What is the result of any number divided by itself? Answer 1. 2.) What happens when you multiply any number by 1? Answer: It does not affect the value. 3.) What happens when you square a square root? Answer, it goes to the og square, you pretty much just skip the middle step Exex. 5⋅5=25=5\sqrt5\cdot\sqrt5=\sqrt{25}=5

The property used for simplifying radical equations with variables is axx\sqrt[x]{a^{x}}