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Evaluate the following using suitable identities. \[{(105)^3}\]

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Last updated date: 05th Mar 2024
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IVSAT 2024
Answer
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Hint: We break the number inside the bracket in simple terms using addition so we can apply an identity to the expansion. We use the identity \[{(a + b)^3} = {a^3} + {b^3} + 3ab(a + b)\]to open the given term.
*\[{(a + b)^3} = {a^3} + {b^3} + 3ab(a + b)\], where ‘a’ and ‘b’ are different numbers.

Complete step-by-step solution:
We have to evaluate \[{(105)^3}\].......................… (1)
The number inside the bracket is 105.
We can break the number 105 as \[105 = 100 + 5\]
Substitute the value of \[105 = 100 + 5\]in equation (1)
\[ \Rightarrow {(105)^3} = {(100 + 5)^3}\]
Now we use the identity \[{(a + b)^3} = {a^3} + {b^3} + 3ab(a + b)\]to open the RHS of the equation.
Here \[a = 100,b = 5\]
\[ \Rightarrow {(105)^3} = {(100)^3} + {(5)^3} + 3 \times (100) \times (5) \times (100 + 5)\]
We use \[{x^n} = \underbrace {x \times x \times x.....x}_n\], where \[n = 3\]
\[ \Rightarrow {(105)^3} = 100 \times 100 \times 100 + 5 \times 5 \times 5 + 3 \times (100) \times (5) \times (100 + 5)\]
Multiply the required values
\[ \Rightarrow {(105)^3} = 1000000 + 125 + 1500 \times 105\]
\[ \Rightarrow {(105)^3} = 1000000 + 125 + 157500\]
Add the terms in RHS
\[ \Rightarrow {(105)^3} = 1157625\]

\[\therefore \]Value of \[{(105)^3}\] is 1157625

Note: Students are likely to make the mistake of calculating \[{(105)^3} = 105 \times 105 \times 105\]but we have to find the value using identities, so we can use different identities after breaking up the terms. Students are advised not to use calculator for calculating the direct value of \[{(105)^3}\]
Alternate Method:
We have to evaluate \[{(105)^3}\]
We use the law of exponents \[{x^n} = {x^{n - 1}} \times {x^1}\] to open the power of the number
Since we can write \[3 = 2 + 1\]
\[ \Rightarrow {(105)^3} = {(105)^{2 + 1}}\]
Now we use the law of exponents to break the RHS where base is the same and powers are added.
\[ \Rightarrow {(105)^3} = {(105)^2} \times (105)\]..................… (2)
The number inside the bracket is 105.
We can break the number 105 as \[105 = 100 + 5\]
Substitute the value of \[105 = 100 + 5\] in first bracket in equation (2)
\[ \Rightarrow {(105)^3} = {(100 + 5)^2} \times (105)\]
Now we use the identity \[{(a + b)^2} = {a^2} + {b^2} + 2ab\] to open the first bracket in the RHS of the equation.
Here \[a = 100,b = 5\]
\[ \Rightarrow {(105)^3} = \left\{ {{{(100)}^2} + {{(5)}^2} + 2 \times (100) \times (5)} \right\} \times 105\]
We use \[{x^n} = \underbrace {x \times x \times x.....x}_n\], where \[n = 2\]
\[ \Rightarrow {(105)^3} = \left\{ {100 \times 100 + 5 \times 5 + 2 \times 100 \times 5} \right\} \times 105\]
Multiply the required values
\[ \Rightarrow {(105)^3} = \left\{ {10000 + 25 + 1000} \right\} \times 105\]
Add the terms in bracket in RHS of the equation
\[ \Rightarrow {(105)^3} = \left\{ {11025} \right\} \times 105\]
Multiply the terms in RHS
\[ \Rightarrow {(105)^3} = 1157625\]
 \[\therefore \]Value of \[{(105)^3}\] is 1157625
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