Question

Evaluate ${{\left( 34 \right)}^{2}}$ by using the identity ${{\left( a+b \right)}^{2}}={{a}^{2}}+2ab+{{b}^{2}}$.

Answer 1

Hint: Break the given number 34 into two parts by writing it as $\left( 34 \right)=\left( 30+4 \right)$. Now, apply the formula provided, ${{\left( a+b \right)}^{2}}={{a}^{2}}+2ab+{{b}^{2}}$ by comparing 30 and 4 with $a$ and $b$ respectively. Substitute $a=30$ and $b=4$ in both sides of the given algebraic identity to get the answer.

Complete step-by-step solution:
Here, we have been provided with the expression ${{\left( 34 \right)}^{2}}$ and we have been asked to evaluate it using the provided algebraic identity ${{\left( a+b \right)}^{2}}={{a}^{2}}+2ab+{{b}^{2}}$. But first let us see the definition of algebraic identity.
In mathematics, an identity is equality relating one mathematical expression A to another mathematical expression B, such that A and B produce the same value for all the values of the variables. There are certain types of identities and one of them is algebraic identities. Identities such as ${{\left( a+b \right)}^{2}}={{a}^{2}}+2ab+{{b}^{2}},{{\left( a-b \right)}^{2}}={{a}^{2}}-2ab+{{b}^{2}}$ can be useful in simplifying algebraic expressions and expanding them.
Now, let us come to the question. We need to break 34 into two parts to apply the given identity. So, there are many ways to break 34 like : $\left( 30+4 \right),\left( 31+3 \right),\left( 32+2 \right)$ and so on. But we can see that we can easily find the squares of 30 and 4, so we can break 34 as,
$\left( 34 \right)=\left( 30+4 \right)$
On squaring both sides, we get,
$\Rightarrow {{\left( 34 \right)}^{2}}={{\left( 30+4 \right)}^{2}}$
Now applying the identity ${{\left( a+b \right)}^{2}}={{a}^{2}}+2ab+{{b}^{2}}$by considering $a=30$ and $b=4$, we get,
$\begin{align}
  & \Rightarrow {{\left( 34 \right)}^{2}}={{\left( 30 \right)}^{2}}+{{4}^{2}}+2\times 30\times 4 \\
 & \Rightarrow {{\left( 34 \right)}^{2}}=900+16+240 \\
 & \Rightarrow {{\left( 34 \right)}^{2}}=1156 \\
\end{align}$
Hence our answer is 1156.

Note: One may note that we can also break the term 34 as 34 = 40 - 6 but then we needed to apply ${{\left( a-b \right)}^{2}}$ formula which would be a wrong procedure according to the given question. Note that we can also substitute $b=30$ and $a=4$ as it will not change the result. You may check the answer by multiplying 34 two times.