Question

Evaluate each of the following using identities:
\[{\left( {399} \right)^2}\]
\[{\left( {0.98} \right)^2}\]
\[991 \times 1009\]
A) i) \[59876\]
    ii) \[0.91\]
  iii) \[876590\]
 B) i) \[135879\]
    ii) \[0.87\]
  iii) \[896750\]
C) i) \[159201\]
    ii) \[0.9604\]
  iii) \[999919\]
D) i) \[138760\]
    ii) \[0.9\]
  iii) \[999999\]

Answer 1

Hint: Here in this question we have to write the given number in the form of algebraic identities, we use the formulas of algebraic identities and on simplifying the terms we obtain the required solution for the given question. Here we use the two algebraic identities and they are \[{(a - b)^2} = {a^2} + {b^2} - 2ab\] and \[(a + b)(a - b) = {a^2} - {b^2}\]

Complete step-by-step answer:
The algebraic expression is a combination of the constants, variables and it includes the arithmetic operation symbols. In algebraic expression we have three different kinds namely, monomial, binomial and polynomial expressions.
Here the question is not in the form of algebraic expression but we will write the given number into the algebraic identities form.
We have some algebraic identities for some algebraic expression, but here we write the algebraic identities for the numerals with the help of addition and subtraction arithmetic operations.
Now consider the given expression which is present in the question
we will solve one by one
i) \[{\left( {399} \right)^2}\]
The number 399 can be written as
\[ \Rightarrow 399 = 400 - 1\]
Therefore we have
\[ \Rightarrow {\left( {400 - 1} \right)^2}\]
The above term is in the form of \[{(a - b)^2} = {a^2} + {b^2} - 2ab\], using the formula we have
\[ \Rightarrow {(400 - 1)^2} = {(400)^2} + {(1)^2} - 2(400)(1)\]
On simplifying we have
\[ \Rightarrow 160000 + 1 - 800\]
\[ \Rightarrow 159201\]----(1)
i) \[{\left( {0.98} \right)^2}\]
The number 0.98 can be written as
\[ \Rightarrow 0.98 = 1 - 0.02\]
Therefore we have
\[ \Rightarrow {\left( {1 - 0.02} \right)^2}\]
The above term is in the form of \[{(a - b)^2} = {a^2} + {b^2} - 2ab\], using the formula we have
\[ \Rightarrow {(1 - 0.02)^2} = {(1)^2} + {(0.02)^2} - 2(1)(0.02)\]
On simplifying we have
\[ \Rightarrow 1 + 0.0004 - 0.04\]
\[ \Rightarrow 0.9604\]----(2)
i) \[991 \times 1009\]
The number 991 and 1009 can be written as
\[ \Rightarrow 991 = 1000 - 9\]
\[ \Rightarrow 1009 = 1000 + 9\]
Therefore we have
\[ \Rightarrow \left( {1000 - 9} \right) \times \left( {1000 + 9} \right)\]
The above term is in the form of \[(a + b)(a - b) = {a^2} - {b^2}\], using the formula we have
\[ \Rightarrow (1000 - 9)\left( {1000 + 9} \right) = {(1000)^2} - {(9)^2}\]
On simplifying we have
\[ \Rightarrow 1000000 - 81\]
\[ \Rightarrow 999919\]----(3)
The equations (1), (2) and (3) are obtained answers. Hence the option C) is also the same as the answer which we had obtained.

So, the correct answer is “Option C”.

Note: Since the question contains the algebraic expression and formula of the algebraic identities, it is easy to solve the problem if we knew the standard algebraic formulas. We should take care of signs because it may sometimes be while solving. The like terms can be cancelled or added but not unlike terms.