Question

How do simplify $ {{\left( 2x-3 \right)}^{3}}? $ .

Answer 1

Hint: We will first recall the concept of multiplication of algebraic terms and then we will use the concept that cube of anything is multiple of the same terms taking the terms three times. So, we can write $ {{\left( 2x-3 \right)}^{3}} $ as $ \left( 2x-3 \right)\left( 2x-3 \right)\left( 2x-3 \right) $ and at first we will multiply the first two terms and then the term obtained after multiplication is again multiplied with the same terms.

Complete step by step answer:
We will use the concept of algebraic multiplication to solve the above question.
Since, we have to simplify $ {{\left( 2x-3 \right)}^{3}} $ and we can see that it is the cube of $ \left( 2x-3 \right) $ so, we can write
  $ {{\left( 2x-3 \right)}^{3}} $ as:
 $ \Rightarrow {{\left( 2x-3 \right)}^{3}}=\left( 2x-3 \right)\left( 2x-3 \right)\left( 2x-3 \right) $
Now, we will take first $ \left( 2x-3 \right) $ inside second $ \left( 2x-3 \right) $ and then we will multiply them by opening the brackets.
So, after multiplying the first two terms we will get:
 $ {{\left( 2x-3 \right)}^{3}}=\left( 2x\left( 2x-3 \right)-3\left( 2x-3 \right) \right)\left( 2x-3 \right) $
Now, we will take 2x and -3 inside the bracket, then we will get:
 $ \Rightarrow {{\left( 2x-3 \right)}^{3}}=\left( 4{{x}^{2}}-6x-6x+9 \right)\left( 2x-3 \right) $
After simplifying the above expression, we will get:
 $ \Rightarrow {{\left( 2x-3 \right)}^{3}}=\left( 4{{x}^{2}}-12x+9 \right)\left( 2x-3 \right) $
Now, we \[\left( 2x-3 \right)\] inside the bracket and multiply it with each term and then we will get:
 $ \Rightarrow {{\left( 2x-3 \right)}^{3}}=\left( 4{{x}^{2}}\left( 2x-3 \right)-12x\left( 2x-3 \right)+9\left( 2x-3 \right) \right) $
So, after taking $ 4{{x}^{2}} $ , $ -12x $ , and 9 inside the bracket and multiply with term, then we will get:
 $ \Rightarrow {{\left( 2x-3 \right)}^{3}}=\left( 4{{x}^{2}}\times 2x-4{{x}^{2}}\times 3-12x\times 2x+12x\times 3+9\times 2x-3\times 9 \right) $
Now, after simplifying we will get:
 $ \Rightarrow {{\left( 2x-3 \right)}^{3}}=\left( 8{{x}^{3}}-12{{x}^{2}}-24{{x}^{2}}+36x+18x-27 \right) $
 $ \Rightarrow {{\left( 2x-3 \right)}^{3}}=\left( 8{{x}^{3}}-36{{x}^{2}}+54x-27 \right) $
Hence, the simplified form of $ {{\left( 2x-3 \right)}^{3}} $ is given as $ \left( 8{{x}^{3}}-36{{x}^{2}}+54x-27 \right) $ .
This is our required solution.



Note:
A student is required to note that when we multiply two algebraic expressions for example: Let us multiply two algebraic expressions $ \left( x-5 \right) $ and $ \left( y-7 \right) $ , so we can multiply both of them i.e. $ \left( x-5 \right)\times \left( y-7 \right) $ by taking each term of the first bracket and then multiplying them with the second bracket, then we will get:
 $ \Rightarrow \left( x-5 \right)\times \left( y-7 \right)=x\left( y-7 \right)-5\left( y-7 \right) $
Now, we will expand the above result, then we will get:
 $ \Rightarrow xy-7x-5y+35 $
We can also solve the given expansion by using the expansion of $ {{\left( a+b \right)}^{3}}={{a}^{3}}+{{b}^{3}}+3ab\left( a+b \right) $ to simplify $ {{\left( 2x-3 \right)}^{3}} $ . Here a is equivalent to 2x and b is equivalent to -3 so we will put the value of a and b in the above expansion. This is our required way of solution.