Question

Using identities, evaluate: $78\times 82$.

Answer 1

Hint: In this question, we will use identity $\left( a+b \right)\left( a-b \right)={{a}^{2}}-{{b}^{2}}$ and identity ${{\left( a\times b \right)}^{m}}={{a}^{m}}\times {{b}^{m}}$ to evaluate given expression with less calculations.

Complete step-by-step answer:
In this question we are to find the product of two number 78 and 82 using identities of algebra.
We have expression, $78\times 82$
We try to write numbers which end with zeroes to make our calculation easy.
So, we can write 78 as the difference of 80 and 2.
That is $78=80-2$.
Also, we can write 82 as the sum of 80 and 2.
That is $82=80+2$.
Using these values of 78 and 82 in given expression, we get,
$\begin{align}
  & 78\times 82 = \left( 80-2 \right)\times \left( 80+2 \right) \\
\end{align}$
Now, we have an identity in algebra, which is given as,
$\left( a+b \right)\left( a-b \right)={{a}^{2}}-{{b}^{2}}$.
Where a and b are any real numbers.
Using this identity in our expression we can write,
$78\times 82=\left( {{80}^{2}}-{{2}^{2}} \right)$
Now, we can write, 80 as product of 8 and 10,
That is $80=8\times 10$.
So, from above expression, we get,
$78 \times 80={{\left( 8\times 10 \right)}^{2}}-{{2}^{2}}$
Now, we have an identity in algebra which is given as,
${{\left( a\times b \right)}^{m}}={{a}^{m}}\times {{b}^{m}}$
Where a, b and m are real numbers.
Using this identity in our expression, we can write,
$78\times 82={{8}^{2}}\times {{10}^{2}}-{{2}^{2}}$
Writing values of square, we get,
\[\begin{align}
  & 78\times 82=64\times 100-4 \\
 & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,=6400-4 \\
\end{align}\]
Subtracting, we get,
$78\times 82=6396$.
Hence, the value of the given expression is 6396.

Note: In this question, we can also use other identities, but always try to choose those identities where numbers will end with zero to make evaluation simple.