Square and Cube Roots – Concept & Explanation

Key Concepts

• Square and cube

• Square root and cube root

• Perfect squares and perfect cubes

• Evaluate square roots and cube roots to solve problems

Evaluate square roots and cube roots

Square: 

To square, a number, multiply it by itself. 

For example, 4 squared equals 16. 

Square root: 

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

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

Perfect squares: 

A perfect square is a number whose square root is an integer. 

The following table shows the list of perfect squares: 

1.4.1 Evaluate Cube roots to Solve Problems 

What is a cube root? 

The formula of cube root is a = ∛b, where a is the cube root of b. For example, the cube root of 125 is 5 because 5 × 5 × 5 = 125. 

Example 1: Calculate the cube root of 343. 

Solution: 

To find the cube root of 343 

Use the cube root formula, 

Cube root of 343 = ∛343=∛ (7×7×7) 

= 7 

1.4.2 Evaluate Perfect Squares and Perfect Cubes 

Perfect squares are numbers whose square roots are whole numbers. 

Diagrammatic representation of perfect squares:  

Perfect squares 

These numbers are perfect squares because their square roots are whole numbers. 

Perfect squares have the easiest square roots to find because they are whole numbers. 

What is the square root of 5? 

First, the diagram shows that 4, 5, and 9 are all “perfectly square,” meaning that in each square, the sides are equal. 

Next, the square root of 5 cannot be easily determined because 5 does not appear in the multiplication tables. You cannot easily identify a number to multiply times itself to get 5 as the answer.  

Then, you know that 5 is not a perfect square. Its square root is not a whole number but is somewhere between 2 and 3. It falls between the perfect squares of 4 and 9. 

What is perfect square or imperfect square? 

How to find the perfect cube? 

The perfect cube of a number can be checked by following the steps given below: 

Step 1: Prime factorize the given number starting from the smallest prime number (2). 

Step 2: Once the prime factorization is done, club every three same factors together. 

Step 3: Repeat the step for all the sets of the group of the same three factors. If there are any factors that are left behind and do not fit into a group of three same factors, then the given number is not a perfect cube. Otherwise, the given number is a perfect cube. 

Perfect Cube Example 

1.4.3 Evaluate Square roots to Solve Problems 

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 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. 

S = √A, and A = 225 

S=√225 

Step 5: Solve the equation using good algebra techniques. Round to one decimal place. 

S = 15 

Step 6: Answer the question with a complete sentence.  

Each side of the patio should be 15 feet.  

Exercise:

1. Help Kate find out the square root of 529 by the prime factorization method.
2. Find the square root of √121, and identify whether it is perfect square or not.
3. Find the square root of √74, and identify whether it is perfect square or not.
4. Find the cubes of

         i. 11                 ii. 12              iii. 21

• Emily’s father’s age is 27 years. Find the age of Emily if her age is the cube root of her father’s age.
• Find the smallest number by which 1944 must be multiplied so that the product is a perfect cube.
• Find the volume of a cube, one face of which has an area of 64m²
• Which of the following numbers are not perfect cubes?

         i. 64      ii. 216        iii. 243         iv. 1728

• Find the cube root of each of the following natural numbers

        i. 343      ii. 2744       iii. 4913        iv. 1728

A school auditorium has 3136 total number of seats, if the number of seats in the row is equal to the number of seats in the columns. Calculate the total number of seats in a row

What we have learned:

• 1.3.1 About fractions, rational numbers, decimal numbers, irrational numbers, number line, perfect squares and approximation of irrational numbers

• 1.3.2 Comparing two irrational numbers

• 1.3.3 Comparing and ordering rational and irrational numbers

Concept Map :