Answer
405k+ views
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
*\[{(a + b)^3} = {a^3} + {b^3} + 3ab(a + b)\], 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
Recently Updated Pages
How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Why Are Noble Gases NonReactive class 11 chemistry CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let X and Y be the sets of all positive divisors of class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let x and y be 2 real numbers which satisfy the equations class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let x 4log 2sqrt 9k 1 + 7 and y dfrac132log 2sqrt5 class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let x22ax+b20 and x22bx+a20 be two equations Then the class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Trending doubts
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
At which age domestication of animals started A Neolithic class 11 social science CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Which are the Top 10 Largest Countries of the World?
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Give 10 examples for herbs , shrubs , climbers , creepers
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Difference Between Plant Cell and Animal Cell
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Write a letter to the principal requesting him to grant class 10 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Change the following sentences into negative and interrogative class 10 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Fill in the blanks A 1 lakh ten thousand B 1 million class 9 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)