Square Root ((Reference))

The square root of a positive number x x is a number such that when it is squared, the number x x results.

Every positive number has two square roots, one positive and one negative. They are opposites of each other.

Principal Square Root x x ((Reference))

If x x is a positive real number, then

x x represents the positive square root of x x . The positive square root of a number is called the principal square root of the number.

Secondary Square Root − x − x ((Reference))

− x − x represents the negative square root of x x . The negative square root of a number is called the secondary square root of the number.

Radical Sign, Radicand; and Radical ((Reference))

In the expression x , x ,
√ is called the radical sign.
x x is called the radicand.
x x is called a radical.

The horizontal bar that appears attached to the radical sign, √, is a grouping symbol that specifies the radicand.

Meaningful Expressions ((Reference))

A radical expression will only be meaningful if the radicand (the expression under the radical sign) is not negative:

−25 −25 is not meaningful    and     −25 −25 is not a real number

Simplifying Square Root Expressions ((Reference))

If a a is a nonnegative number, then

a 2 =a a 2 =a

Perfect Squares ((Reference))

Real numbers that are squares of rational numbers are called perfect squares.

Irrational Numbers ((Reference))

Any indicated square root whose radicand is not a perfect square is an irrational number.

2 , 5 2 , 5   and   10 10 are irrational numbers

The Product Property ((Reference))

xy = x y xy = x y

The Quotient Property ((Reference))

x y = x y y≠0 x y = x y y≠0

Be Careful ((Reference))

x+y ≠ x + y ( 16+9 ≠ 16 + 9 ) x−y ≠ x − y ( 25−16 ≠ 25 − 16 ) x+y ≠ x + y ( 16+9 ≠ 16 + 9 ) x−y ≠ x − y ( 25−16 ≠ 25 − 16 )

Simplified Form ((Reference))

A square root that does not involve fractions is in simplified form if there are no perfect squares in the radicand.

A square root involving a fraction is in simplified form if there are no

Rationalizing the Denominator ((Reference))

The process of eliminating radicals from the denominator is called rationalizing the denominator.

Multiplying Square Root Expressions ((Reference))

The product of the square roots is the square root of the product.

x y = xy x y = xy

Dividing Square Root Expressions ((Reference))

The quotient of the square roots is the square root of the quotient.

x y = x y x y = x y

Addition and Subtraction of Square Root Expressions ((Reference))

a x +b x =( a+b ) x a x −b x =( a−b ) x a x +b x =( a+b ) x a x −b x =( a−b ) x

Square Root Equation ((Reference))

A square root equation is an equation that contains a variable under a square root radical sign.

Solving Square Root Equations ((Reference))