Kerala Board Solutions for Chapter: Identities, Exercise 4: Exercise 4

Is there a general method to compute the squares of numbers like $1\frac{1}{2},2\frac{1}{2},3\frac{1}{2}\dots ?$ Explain it using algebra.

Given below is a method to calculate ${37}^2?$

Check this for some more two-digit numbers.

Given below is a method to calculate ${37}^2$?

$$\begin{gathered} \begin{array}{lll}{3}^2=9& 9\times 100& 900\\ 2\times (3\times 7)=42 & 42\times 10& 420\end{array} \\ \underline{ 7^2 49} \\ {37}^2 1369 \end{gathered}$$

Explain why this is correct, using algebra.

Given below is a method to calculate ${37}^2?$

$\begin{array}{lrr}{3}^2=9& 9\times 100& 900\\ 2\times (3\times 7)=42& 42\times 10& 420\\ 7^2& & 49\\ {37}^2& & 1369\end{array}$

Find an easy method to compute squares of number ending in $5$.

Look at this pattern,

$1^2+(4\times 2)=3^2$

$2^2+(4\times 3)=4^2$  

$3^2+(4\times 4)=5^2$  

Write the next two lines.

Look at this pattern,

$1^2+(4\times 2)=3^2$

$2^2+(4\times 3)=4^2$  

$3^2+(4\times 4)=5^2$  

Explain the general principle using algebra.

Explain using algebra, the fact that the square of any natural number which is not a multiple of $3$, leaves remainder $1$ on division by $3$ .

Prove that for any natural number ending in $3$, its square ends in $9$. What about numbers ending in $5$? And numbers ending in $4$?