Simplifying Radical Expressions

Algebra 1

If b 2 = a, then b is a square root of a.

Meaning Positive

Square Root

Negative

Square Root

The positive and negative square

roots

Symbol

Example

39 39 39

Simplifying Radical Expressions

100 4 25 36 4 9

326

Product Property for Radicals

ab a b

10 2 5 36 4 9

Simplifying Radical Expressions

• A radical has been simplified when its radicand contains no perfect square factors.

• Test to see if it can be divided by 4, then 9, then 25, then 49, etc.

• Sometimes factoring the radicand using the “tree” is helpful.

Product Property for Radicals

50 25 2 5 2

14 7x x

Steps

1. Try to divide the radicand into a perfect square for numbers

2. If there is an exponent make it even by using rules of exponents

3. Separate the factors to its own square root

4. Simplify

Simplify: 45

59

53

Simplify: x49

x49

x7

Simplify: t52

t134

t132

Simplify:12x

6x

26x

Square root of a variable to an even power = the

variable to one-half the power.

Simplify:88y

44y

Square root of a variable to an even power = the

variable to one-half the power.

Simplify:13x

xx6

12x x

112xx

Simplify:27x

13x x

26x x

Simplify:836x

46x

2 86 x

Simplify:1049y

57 y

Simplify:645x

33 5x

69 5x

Simplify:750y

35 2y y

625 2y y

Simplify:948y

Simplify:1680y

84 5y

1616 5y

Simplify:336x

6x x

Simplify:4 7448x y

2 38 7x y y

4 6 1124x y y

2 32x y 4 28y

2 32 2x y 4 7 y

• square root: one of two equal factors of a given number. The radicand is like the “area” of a square and the simplified answer is the length of the side of the squares.

• Principal square root: the positive square root of a number; the principal square root of 9 is 3.

• negative square root: the negative square root of 9 is –3 and is shown like

• radical: the symbol which is read “the square root of a” is called a radical.

• radicand: the number or expression inside a radical symbol --- 3 is the radicand.

• perfect square: a number that is the square of an integer. 1, 4, 9, 16, 25, 36, etc… are perfect squares.

• irrational number: a number whose decimal form is nonterminating and nonrepeating.

• Rational number: a number that can be written in the form a/b, where a and b are integers (b cannot equal 0)

• radical expression: an expression that contains a radical.

39

39

3

2

25 4

Simplify: 363y

183y

Simplify: 50202 2 xx

52 x

25102 2 xx

252 x

Simplify:2 6 9y y

3y

23y

Simplify: 104 5x

52 5x

104 5x

Simplify: 950 7y

45 7 2 7y y

825 7 2 7y y