Uses of Square and Cube roots

Key Concepts

1. Solving Equations involving Perfect Squares

2. Solving Equations involving Perfect cubes

3. Solving Equations involving Imperfect Squares and Cubes

Solve Equations Using Square Roots and Cube Roots 

Introduction:

• In this chapter, we will learn to solve equations involving perfect squares and cubes. 

• Solve equations involving imperfect squares and imperfect cubes. 

Square root: 

The square root of a number is the number that gets multiplied by itself to give the product. 

The square root of 9 is 3, because when 3 is multiplied by itself we get 9. 

Perfect squares: 

Perfect squares are numbers whose square roots are whole numbers. 

Diagrammatic representation of perfect squares:  

What is a Cube Root? 

The cube root of a number is a special value that when cubed gives the original number. 

This is called 3 Cubed i.e., 3 × 3 × 3 

So, cube root of 27 is 3. 

Perfect Cubes: 

Perfect cubes are numbers whose cube roots are whole numbers. 

5 is the cube root of 125. 

Since 5 is a whole number, 27 is called perfect cube. 

1.5 Solve Equations Using Square Roots and Cube Roots: 

1.5.1 Solve equations involving perfect squares 

George wants to make a square patio. He has enough concrete to pave an area of 225 square feet. Use the formula s =√A to find the length of each side of the patio. 

Step 1: Read the problem. Draw a figure of square patio and label it with the given information. 

A = 225 square feet 

Step 2: Identify what you are looking for.  

The length of a side of the square patio. 

Step 3: Name what you are looking for by choosing a variable to represent it. 

Let s = the length of a side. 

Step 4: Translate into an equation by writing the appropriate formula or model for the situation. 

Substitute the given information. 

A = s², and A = 225 

225 = s²  

√225 = √s² 

Step 5: Solve the equation using good algebra techniques. 

25 = s 

Step 6: Answer the question with a complete sentence.  

Each side of the patio should be 25 feet. 

1.5.2 Solve equations involving perfect cubes 

George constructed a cubical water tank that has base area of 49 m² and is

13

water filled. Let’s find the amount of water it can hold. 

A cube has all three sides equal. 

Let the side be a 

so base area = a² = 49 

a = 7m 

So, volume of water it can hold = a × a × a 

 = 7 × 7 × 7 

 = 343m³  

1313

of the tank is already filled. So, remaining part of the tank is

2323

343 ×

2323

= 228.6 

1m³ = 1000L 

So, the amount of water it can hold = 228.6 × 1000 = 228600 L 

1.5.3 Solve equations involving imperfect squares and cubes 

Let’s solve for x in the equation x ² = 27 

Step1: Apply root on both the sides for simplification 

√x ² = √27 

Step2: Simplify the equation 

x = √27 

x = +√27 or -√27 

Let’s solve for x in the equation x ³ = 11 

Step1: Apply cube root on both the sides for simplification 

3√x ³ = 3√11 

Step2: Simplify the equation 

x = 3√11 

Exercise:

1.     Find the square of the following numbers.

         (i)    32                        (ii)     35                           (iii)   86                      

         (iv) 93                        (v)     71                           (vi)    46

2.     Find which of the following numbers are perfect squares?

         (i)    225                     (ii)     189                        (iii)   441                    

         (iv) 729                     (v)     1575                      (iv)    900

3.     Which of the following numbers are not perfect cubes?

         (i)    216                     (ii)     128                        (iii)   1000                  (iv)    100         (v)                       46656

4.     Which of the following are perfect cubes?

         (i)    400                     (ii)     3375                     (iii)   8000                  (iv)    15625

         (v)   9000                  (vi)    6859                     (vii)  2025                  (viii) 10648

5.     Is 392 a perfect cube? If not, find the smallest natural number by which 392 must be multiplied so that the product is a perfect cube.

6.     Find the smallest number by which each of the following numbers must be multiplied to obtain a perfect cube.

         (i)    243                     (ii)     256                        (iii)   72                        (iv)    675         (v)                       100

7.     Find the smallest number by which each of the following numbers must be divided to obtain a perfect cube.

         (i)    81                        (ii)     128                        (iii)   135                     (iv)    192         (v)                       704

8.     Parikshit makes a cuboid of plasticine of sides 5 cm, 2 cm, 5 cm. How many such cuboids will he need to form a cube?

9.     Find the cube root of each of the following numbers by prime factorization method.

         (i)    64                        (ii)     512                        (iii)   10648                (iv)    27000

10.  The students of Class VIII of a school donated 2401 dollars in all, for a cyclone relief fund. Each student donated as many dollars as the number of students in the class. Find the number of students in the class.

What have we learned:

• Square roots and cube roots.

• Perfect squares and perfect cubes.

• How to use perfect squares and perfect cubes to solve equations.

• How to use imperfect squares and imperfect cubes to solve equations.

Concept Map: