Square RootsName: Period:

A root (also known as a radical) is the inverse operation of applying an exponent. Any exponent can be reversed by using a radical.

Parts of a root

You do not need to write the index for a square root, it is assumed to be 2.

ExamplesExponent

Verbal ExpressionExponentNumerical example

Inverse(Reverse)

RootVerbal Expression

RootNumerical example

Squaring a number(raising to the 2nd power)

42=16 ⇔ The square root of a number

√16=4Squaring a number

(raising to the 2nd power)82=64 ⇔ The square root of

a number√64=8

Squaring a number(raising to the 2nd power)

122=144 ⇔ The square root of a number

√144=12Squaring a number

(raising to the 2nd power)92=81 ⇔ The square root of

a number√81=9

A perfect root is found by raising an integer to whole number exponent.

Examples

Symbolic Models

Practice: Find the square root of the following numbers. Identify each root as rational or irrational.

1. √64 2. √121 3. √225 4. −√25

A non-perfect root is found by raising a non-integer to a whole number exponent. A non-perfect square root can be rational or irrational. Since we cannot find the exact value of the root, we will estimate what two consecutive integers the root lies between.

Steps for estimating non-perfect square roots

1 Write a list of perfect squares (See list to the right.)2 Determine what two values the radicand is in between.3 Determine the corresponding roots the square root would be

in between.

Example: Find the √551 Write a list of perfect squares See chart above2 Determine what two values the radicand is in between. The radicand in this example is

55. 55 is between the values 49 and 64.

3 Determine the corresponding roots the square root would be in between.

√49=7 and √64=8 ,

therefore √55 is between 7 and 8.

Practice: Estimate what two consecutive integers the square root of the following numbers is between. Identify each root as rational or irrational.

5. √72 6. √103 7. −√136 8. √19