Square Roots and Cubes

What is Square root?

We know that:-

    \(2^{2}=2 \times 2=4\)

    \(5^{2}=5 \times 5=25\)

We say that a square of 2 is 4, the square of 5 is 25. In other words, we say that the square root of 4 is 2; the square root of 25
is 5.

we write as: 

      \(\begin{aligned} \sqrt{4} &=2 \\ \sqrt{25} &=5 \end{aligned}\)

Perfect Square - A natural number is called a perfect square of any natural number. Thus 4,9,16,25,36 etc. are perfect squares.

Note

  1. The square root of a negative rational number is not possible.
  2. The square root of a natural number can be a rational number or an irrational number."
  3. If a and b are a natural number and perfect squares.

                                    Then \(\quad \sqrt{a \times b}=\sqrt{a}\times \sqrt{b}\)

                                     and    \(\sqrt{\frac{a}{b}}=\frac{\sqrt{a}}{\sqrt{b}}\)

 

Remember Points in Square

Remember Points: -

  1.  Square of an even number is always even.
  2.  Square of an odd number is always odd.
  3.  Square of all natural numbers ends in 0,1,4,5,6 or 9. Numbers ending in the digits 2,3,7, and 8 are never perfect squares.
  4. Numbers ending in the digits 2,3,7 and 8 are never perfect squares.
  5. For three natural numbers a, b, c if then \(a^{2}+b^{2}=c^{2},\) (a, b, c) are called Pythagorean Triplet.
    (For example;\(3^{2}+4^{2}=5^{2} \)\( 5^{2}+12^{2}=13^{2}\), the number (3,4,5) and (5,12,13) are Pythagorean Triplets.   
  6. The square root of natural numbers can find out either by factor method or by long division method. The square root of a positive rational number-We know
    •          \( \begin{aligned}\left(\frac{2}{5}\right)^{2} &=\frac{2}{5} \times \frac{2}{5}=\frac{4}{25} \\\left(\frac{8}{13}\right)^{2} &=\frac{8}{13} \times \frac{8}{13}=\frac{64}{169} \\(0 \cdot 3)^{2} &=0 \cdot 3 \times 0 \cdot 3=0-09 \\(0 \cdot 25)^{2} &=0.25 \times 0.25=0 \cdot 0625 \end{aligned}\)
    • thus, \(\begin{aligned} \sqrt{\frac{4}{25}} &=: \frac{2}{5} \\ \sqrt{\frac{64}{169}} &=\frac{8}{13} \\ \sqrt{0.0625} &=0.25 \end{aligned}\)
  7. Here we observe that square root of a rational number less than 1 is always greater than the rational number. \(\frac{2}{5}>\frac{4}{25} : 0.25>0.0625\)

            In case of a natural number square root is always less than the number except 1 whose square root is itself.