IESN - Square and Cube Roots Lesson

Radicals

In this lesson, we will learn how to represent solutions to equations using radicals (square and cube root symbols).  We will learn that all perfect squares have two roots and will be able to evaluate square roots of perfect squares up to 625, and cube roots of perfect cubes up to 1000.  We will also learn to estimate non-perfect squares(irrational numbers) to the tenths place.

Perfect Squares and Cubes

Squares and square roots are important concepts in all types of math.  One important feature of square roots is that they can be used to find the side length of a square when you know the area.  A number that is multiplied by itself to form a product is called a square root of the product.  The operations of squaring and finding a square root are inverse operations. 

Cubes and cube roots are also used in different types of math.  Cube roots can be used to find the side length of a cube when you know the volume.  A number that used as a factor three times (2 x 2 x 2) is called a cube root of the product.  The operations of cubing and finding a cube root are inverse operations.

 

Below are the perfect squares that you will need to learn.  The primary roots are listed (positive), but negative integers squared will give you the same perfect square.

Below are the perfect cubes that you will need to learn.  The cube  roots have to be positive unless you are looking for the negative cube root.

The radicals can be part of an equation and can also be the solution to an equation.

Example

$4\sqrt{25}=4\times5=20$

$3\sqrt{4}+\sqrt{16}=3\times2+4=10$

A square has side lengths of 6 - The area is $\sqrt{36}$

A cube has side lengths of 3 - The volume is $\sqrt[3]{27}$

Estimating Roots for Non-perfect Squares

Most numbers are not perfect squares; therefore, we must estimate.  The first step is to decide between which two perfect squares the non-perfect square lies between.  Then we will determine the decimal component.

All of the numbers between the perfect squares like: $\sqrt{2},\sqrt{3},\sqrt{5},\sqrt{6},\sqrt{7,}\sqrt{8}$ must be estimated.

These non perfect squares fall between the roots of the perfect squares on the number line.  Therefore, we write them as decimals by approximating.

Example and steps to estimate the square root of 20

Estimation Example

Without using a calculator, estimate $\sqrt{14}$ . The square root will be between $\sqrt{9}and\sqrt{16}$, so the whole number will be between 3 and 4.

The difference of the perfect squares is 7(denominator) the difference between the first two numbers 9 and 14 is 5 (numerator).  The estimated root for $\sqrt{14}=3\frac{5}{7}$ .  Turn this into a decimal and you get 3.71.

Practice

If you would like to practice these types of problems and check your work before you complete your graded homework assignment, click here. Links to an external site.  Make sure you check your answers and look at the course resources or ask your teacher if there are any problems you do not understand. Links to an external site.

IMAGES CREATED BY GAVS (INSERT CREDIT FOR IMAGES OR CONTENT HERE IN ALL CAPS!)