Question

How do you factor \[27{a^3} - 64{b^3}\] ?

Answer 1

Hint: Here in this question, we have to find the factors of the given equation. If see the equation it is in the form of \[{a^3} - {b^3}\] . We have a standard formula on this algebraic equation and it is given by \[{a^3} + {b^3} = (a - b)({a^2} + ab + {b^2})\] , hence by substituting the value of a and b we find the factors.

Complete step-by-step answer:
The equation is a algebraic equation or expression, where algebraic expression is a combination of variables and constant.
Now consider the given equation \[27{a^3} + 64{b^3}\] , let we write in the form of exponential form. The number \[27{a^3}\] can be written as \[3a \times 3a \times 3a\] and the \[64{b^3}\] can be written as \[4b \times 4b \times 4b\] , in the exponential form it is \[{\left( {4b} \right)^3}\] . The number \[27{a^3}\] written as \[3a \times 3a \times 3a\] and in exponential form is \[{\left( {3a} \right)^3}\] . Therefore, the given equation is written as \[{\left( {3a} \right)^3} - {\left( {4b} \right)^3}\] , the equation is in the form of \[{a^3} - {b^3}\] . We have a standard formula on this algebraic equation and it is given by \[{a^3} - {b^3} = (a - b)({a^2} + ab + {b^2})\] , here the value of a is \[3a\] and the value of b is \[4b\] . By substituting these values in the formula, we have
 \[27{a^3} - 64{b^3} = {\left( {3a} \right)^3} - {\left( {4b} \right)^3} = (3a - 4b)({(3a)^2} + (3a)(4b) + {(4b)^2})\]
On simplifying we have
 \[ \Rightarrow 27{a^3} - 64{b^3} = (3a - 4b)(9{a^2} + 12ab + 16{b^2})\]
The second term of the above equation can be solved further by using factorisation or by using the formula \[({a^2} + {b^2}) = {(a + b)^2} - 2ab\]
The above equation is written as
 \[ \Rightarrow 27{a^3} - 64{b^3} = (3a - 4b)(9{a^2} + 16{b^2} + 12ab)\]
So let we consider the second term and solve it so we have
 \[ \Rightarrow (9{a^2} + 16{b^2} + 12ab) = {(3a + 4b)^2} - 2(3a)(4b) + 12ab\]
On simplifying we have
 \[ \Rightarrow (9{a^2} + 16{b^2} + 12ab) = {(3a + 4b)^2} - 24ab + 12ab\]
On further simplification we have
 \[ \Rightarrow (9{a^2} + 16{b^2} + 12ab) = {(3a + 4b)^2} - 12ab\]
If we see the simplification of the second term, it looks like the bilk term. So there is no need to simplify the second term.
Therefore, the factors of \[27{a^3} - 64{b^3}\] is \[(3a - 4b)(9{a^2} + 12ab + 16{b^2})\]
So, the correct answer is “\[(3a - 4b)(9{a^2} + 12ab + 16{b^2})\]”.

Note: To find the factors for algebraic equations or expressions, it depends on the degree of the equation. If the equation contains a square then we have two factors. If the equation contains a cube then we have three factors. Here this equation also contains 3 factors, the two factors may be imaginary.