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Evaluate \[8.5 \times 9.5\] using a suitable standard identity.

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Last updated date: 13th Jun 2024
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Answer
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Hint: Here, we need to find the value of \[8.5 \times 9.5\] using a suitable standard identity. We will rewrite the two numbers as a sum or difference of two numbers in such a way that the product can be found using one of the standard algebraic identities. Then, we will simplify the expression to find the required value.

Formula Used: The product of the sum of two number and the difference of two numbers can be calculated using the algebraic identity \[\left( {a - b} \right)\left( {a + b} \right) = {a^2} - {b^2}\].

Complete step-by-step answer:
We can evaluate the given product using any of the two identities \[\left( {x + a} \right)\left( {x + b} \right) = {x^2} + \left( {a + b} \right)x + ab\] or \[\left( {a - b} \right)\left( {a + b} \right) = {a^2} - {b^2}\].
We will use the second identity to solve this problem.
First, we will rewrite the given numbers as the sum or difference of two numbers such that the identity is applicable.
We know that \[8.5\] is the difference of 9 and \[0.5\], and \[9.5\] is the sum of 9 and \[0.5\].
Therefore, substituting \[8.5 = 9 - 0.5\] and \[9.5 = 9 + 0.5\], we can rewrite the product as
\[ \Rightarrow 8.5 \times 9.5 = \left( {9 - 0.5} \right)\left( {9 + 0.5} \right)\]
Now, we know that the product of the sum of two number and the difference of two numbers can be calculated using the algebraic identity \[\left( {a - b} \right)\left( {a + b} \right) = {a^2} - {b^2}\].
Substituting \[a = 9\] and \[b = 0.5\] in the algebraic identity, we get
\[ \Rightarrow \left( {9 - 0.5} \right)\left( {9 + 0.5} \right) = {9^2} - {\left( {0.5} \right)^2}\]
Simplifying the expression, we get
\[ \Rightarrow 8.5 \times 9.5 = 81 - 0.25\]
Subtracting \[0.25\] from 81, we get
\[ \Rightarrow 8.5 \times 9.5 = 80.75\]
Therefore, the value of the product \[8.5 \times 9.5\] is \[80.75\].

Note: We can also solve the problem using the identity \[\left( {x + a} \right)\left( {x + b} \right) = {x^2} + \left( {a + b} \right)x + ab\].
Substituting \[8.5 = 8 + 0.5\] and \[9.5 = 8 + 1.5\], we can rewrite the product as
\[ \Rightarrow 8.5 \times 9.5 = \left( {8 + 0.5} \right)\left( {8 + 1.5} \right)\]
Now, we will use the algebraic identity \[\left( {x + a} \right)\left( {x + b} \right) = {x^2} + \left( {a + b} \right)x + ab\].
Substituting \[x = 8\],\[a = 0.5\] and \[b = 1.5\] in the algebraic identity, we get
\[ \Rightarrow \left( {8 + 0.5} \right)\left( {8 + 1.5} \right) = {8^2} + \left( {0.5 + 1.5} \right)8 + \left( {0.5} \right)\left( {1.5} \right)\]
Simplifying the expression, we get
\[\begin{array}{l} \Rightarrow 8.5 \times 9.5 = 64 + 2 \times 8 + 0.75\\ \Rightarrow 8.5 \times 9.5 = 64 + 16 + 0.75\end{array}\]
Adding the terms of the expression, we get
\[ \Rightarrow 8.5 \times 9.5 = 80.75\]
\[\therefore\] The value of the product \[8.5 \times 9.5\] is \[80.75\].