Squares and Square Roots

 
3.1  Squares and Square Roots
 
Squares:
 
  Definition  
     
 

A number that indicates a number is multiplied by the number itself.

 
 
  • Examples: \(2^2,\, 9^2,\, 5^2\)
 
Perfect squares:
 
  Defintion  
     
 

A number that is equal to the square of a whole number.

 
 
  • Examples: \(1,\, 4,\, 9\)
 
Determine a number is a perfect square:
 
  • Perfect square can be written as a product of two equal factors.
 
  Example  
     
 

\(\begin{aligned} 144&=12\times12 \\\\&=12^2. \end{aligned}\)

\(144\) is a perfect square.

 
 
Relationship between squares and square roots:
 
  • Finding the square and finding the square root are inverse operations.
 

 
  Example  
     
 

The square of \(8\) is \(64\).

The square root of \(64\) is \(8\).

\(8\times8=64\)

Thus,

\(\begin{aligned} \sqrt{64}&=\sqrt{8\times8} \\\\&=8. \end{aligned}\)

 
 
The square of a number:
 
  Example  
     
 

Calculate:

(i)

\(\begin{aligned} 3^2&=3\times3 \\\\&=9. \end{aligned}\)

(ii)

\(\begin{aligned} \bigg(\dfrac{2}{5}\bigg)^2&=\dfrac{2}{5}\times\dfrac{2}{5} \\\\&=\dfrac{4}{25}. \end{aligned}\)

 
 
The square root of a number:
 
  Example  
     
 

Solve:

(i)

\(\begin{aligned} \sqrt{121}&=\sqrt{11\times11} \\\\&=\sqrt{11^2}\\\\&=11. \end{aligned}\)

(ii)

\(\begin{aligned} \sqrt{\dfrac{25}{49}}&=\sqrt{\dfrac{5}{7}\times\dfrac{5}{7}} \\\\&=\sqrt{\bigg(\dfrac{5}{7}\bigg)^2} \\\\&=\dfrac{5}{7}. \end{aligned}\)

 
 
Generalisation when two square roots are multiplied:
 

Square roots of the same numbers

  • The product of two equal square root numbers is the number itself.
  • \(\sqrt{a}\times\sqrt{a}=a\)
 

Square roots of different numbers

  • The product of two different square root numbers is the square root of the product of the two numbers.
  • \(\sqrt{a}\times\sqrt{b}=\sqrt{ab}\)