Question

The number of terms in the expansion of ${\left[ {{{\left( {a + 3b} \right)}^2} - {{\left( {a - 3b} \right)}^2}} \right]^2}$when simplified is:
A.4
B.5
C.6
D.7
E.8

Answer 1

Hint: We will use the algebraic identities to find the number of terms in the expansion of the given equation like ${(a + b)^2} = ({a^2} + {b^2} + 2ab)$ and ${(a - b)^2} = ({a^2} + {b^2} - 2ab)$.

Complete step-by-step answer:
We are required to find the number of terms in the expansion of the given equation.
Firstly, let us simplify the given equation: ${\left[ {{{\left( {a + 3b} \right)}^2} - {{\left( {a - 3b} \right)}^2}} \right]^2}$
Here, we will use the identities: ${(a + b)^2} = ({a^2} + {b^2} + 2ab)$
and, ${(a - b)^2} = ({a^2} + {b^2} - 2ab)$
We will use these identities in the equation ${\left[ {{{\left( {a + 3b} \right)}^2} - {{\left( {a - 3b} \right)}^2}} \right]^2}$.
From the algebraic identities we can say, by observation, that here a = a and b = 3 b for the given equation.
Hence, using both the algebraic identities in the given equation, we get
$
   \Rightarrow {[({a^2} + 9{b^2} + 6ab) - ({a^2} + 9{b^2} - 6ab)]^2} \\
   \Rightarrow {[{a^2} + 9{b^2} + 6ab - {a^2} - 9{b^2} + 6ab]^2} \\
   \Rightarrow {[12ab]^2} = 144{a^2}{b^2} \\
 $
 Therefore, the number of terms in the expansion of ${\left[ {{{\left( {a + 3b} \right)}^2} - {{\left( {a - 3b} \right)}^2}} \right]^2}$ is 1.
Hence, none of the options is correct.

Note: In such problems, you may get confused with the identities to be used in the expansion or simplification of the given equation.
We used algebraic identities in the above mentioned question. We can define the algebraic identities as: an equality which holds for each and every value of its variables is called an algebraic identity or an identity is an equality relating one mathematical expression with the other.
In other words, we can say that an algebraic identity is basically an equality in which every value of its variables holds true.