Question

Using the formula ${{\left( a+b \right)}^{2}}=\left( {{a}^{2}}+2ab+{{b}^{2}} \right)$ , Evaluate ${{\left( 630 \right)}^{2}}$ .

Answer 1

Hint: To find the value of ${{\left( 630 \right)}^{2}}$ using the given identity, we have to write ${{\left( 630 \right)}^{2}}$ in the form of ${{\left( a+b \right)}^{2}}$ . We have to use two numbers such that when they are added, the result must be 630. We can rewrite ${{\left( 630 \right)}^{2}}$ as ${{\left( 600+30 \right)}^{2}}$ . Now, we have to use the given identity and simplify.

Complete step by step solution:
We have to find the value of ${{\left( 630 \right)}^{2}}$ using the algebraic identity ${{\left( a+b \right)}^{2}}=\left( {{a}^{2}}+2ab+{{b}^{2}} \right)$ . For this we have to rewrite 630 in the form of $\left( a+b \right)$ . Let us write $630=600+30$ .
$\Rightarrow {{\left( 630 \right)}^{2}}={{\left( 600+30 \right)}^{2}}$
This is in the form ${{\left( a+b \right)}^{2}}$ . Now, let us use the identity ${{\left( a+b \right)}^{2}}=\left( {{a}^{2}}+2ab+{{b}^{2}} \right)$ , where $a=600$ and $b=30$ .
$\Rightarrow {{\left( 600+30 \right)}^{2}}={{600}^{2}}+2\times 600\times 30+{{30}^{2}}$
Let us simplify the above expression.
$\Rightarrow {{\left( 600+30 \right)}^{2}}=360000+36000+900$
Now, let us add all the terms.
$\Rightarrow {{\left( 600 \right)}^{2}}={{\left( 600+30 \right)}^{2}}=396900$

Hence, the value of ${{\left( 630 \right)}^{2}}$ is 396900.

Note: Students can also use other combinations of numbers to get 630. For example, they can write 630 as $\left( 500+130 \right)$ or any other combinations. But the better combination is to take the hundredths and tenths form. Let us evaluate ${{\left( 630 \right)}^{2}}$ using other combinations.
Let us write 630 as $\left( 500+130 \right)$ .
$\Rightarrow {{\left( 630 \right)}^{2}}={{\left( 500+130 \right)}^{2}}$
Now, let us use the given identity.
$\Rightarrow {{\left( 500+130 \right)}^{2}}={{500}^{2}}+2\times 500\times 130+{{130}^{2}}$
Now, we have to simplify the above expression.
$\Rightarrow {{\left( 500+130 \right)}^{2}}=250000+130000+16900$
Now, let us add the terms.
$\Rightarrow {{\left( 630 \right)}^{2}}={{\left( 500+130 \right)}^{2}}=396900$
Therefore, any combination can be taken to evaluate ${{\left( 630 \right)}^{2}}$ , but we have to add the numbers to get 630.
Students have to read the question carefully that the identity to be used is of ${{\left( a+b \right)}^{2}}$ not ${{\left( a-b \right)}^{2}}$ . They must know the algebraic identities very thoroughly.