
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.
 \(\sqrt{a}\times\sqrt{b}=\sqrt{ab}\)
