Question

If a = 2 and b = 3, find the value of \[{{a}^{3}}-{{b}^{3}}\].

Answer 1

Hint: To solve this question, we should know the algebraic property, that is \[{{a}^{3}}-{{b}^{3}}=\left( a-b \right)\left( {{a}^{2}}+ab+{{b}^{2}} \right)\]. Also, while finding the value of \[{{a}^{3}}-{{b}^{3}}\], we have to take care of the signs and calculation mistakes so that we can get an error-free answer.

Complete step-by-step answer:

In this question, we have been given that a = 2 and b = 3, and we are asked to find the value of \[{{a}^{3}}-{{b}^{3}}\]. To find the value \[{{a}^{3}}-{{b}^{3}}\], we should know that \[{{a}^{3}}-{{b}^{3}}\] can be expressed as a multiplication of (a – b) and \[\left( {{a}^{2}}+{{b}^{2}}+ab \right)\], that is
\[{{a}^{3}}-{{b}^{3}}=\left( a-b \right)\left( {{a}^{2}}+ab+{{b}^{2}} \right)\]
Now, we know that a = 2 and b = 3. So by putting these values, we will get the expression as
\[{{2}^{3}}-{{3}^{3}}=\left( 2-3 \right)\left( {{2}^{2}}+{{3}^{2}}+2\times 3 \right)\]
Now, we know that \[{{2}^{2}}\] can be written as \[2\times 2\], that is 4 and \[{{3}^{2}}\] can be written as \[3\times 3\] that is 9. So, we will get the expression as
\[{{2}^{3}}-{{3}^{3}}=\left( 2-3 \right)\left( 4+9+2\times 3 \right)\]
Now, we will simplify the expression by applying arithmetic operations. So, we will get,
\[{{2}^{3}}-{{3}^{3}}=\left( -1 \right)\left( 4+9+6 \right)\]
\[{{2}^{3}}-{{3}^{3}}=\left( -1 \right)\left( 19 \right)\]
\[{{2}^{3}}-{{3}^{3}}=-19\]
Hence, we can say that if a = 2 and b = 3, then \[{{a}^{3}}-{{b}^{3}}\] is –19.

Note: We can also do this question by simply finding the value of \[{{a}^{3}}\], that is \[{{2}^{3}}\] which is equal to \[2\times 2\times 2\] or we can say 8 and \[{{b}^{3}}\] that is \[{{3}^{3}}\] which is equal to \[3\times 3\times 3=21\]. And then we can write \[{{a}^{3}}-{{b}^{3}}={{2}^{3}}-{{3}^{3}}\] which is equal to 8 – 27, that is –19.