Question

Factorize: \[{\left( {0.7} \right)^2} - {\left( {0.3} \right)^2}\]

Answer 1

Hint:
We are given with an equation involving two numbers and are asked to calculate the answer of the operation by factoring the equation. Thus, we will first use the appropriate factoring property. Finally, we will plug in the values and evaluate the value of the calculation.

Formulae Used:
${a^2} - {b^2} = (a - b)(a + b)$
Where $a$ and $b$ are the numbers involved in the operation.

Complete step by step solution:
Here,
The given equation is in the form of${a^2} - {b^2}$.
Thus, we will use the formulation of the factorization of the same.
Hence, we will use the formulation
${a^2} - {b^2} = (a - b)(a + b)$
Thus, we get
${(0.7)^2} - {(0.3)^2} = (0.7 - 0.3)(0.7 + 0.3)$
Further, we get
${(0.7)^2} - {(0.3)^2} = (0.4)(1)$
Hence, we get
${(0.7)^2} - {(0.3)^2} = 0.4$
Additional Information: In order to get to the formulation of${a^2} - {b^2}$, we have to go through the formulation of ${(a + b)^2}$ and ${(a - b)^2}$in order to get the gist of the process of formulation.
Now,
We can easily write,
${(a + b)^2} = (a + b)(a + b)$
Applying the distributive law of multiplication and proceeding, we get
${(a + b)^2} = {a^2} + ab + ab + {b^2}$
Combining the like terms here, we get
${(a + b)^2} = {a^2} + 2ab + {b^2}$
Again,
We can write,
${(a - b)^2} = (a - b)(a - b)$
Applying the distributive law of multiplication and proceeding further, we get
${(a - b)^2} = {a^2} - ab - ab + {b^2}$
Further combining the like terms, we get
${(a - b)^2} = {a^2} - 2ab + {b^2}$
Hence, the workflow of the formulation becomes clear to us.
Now,
For proving the formulation${a^2} - {b^2} = (a - b)(a + b)$, we proceed from the right hand side of the formula.
Thus,
We proceed from $(a - b)(a + b)$
Now,
We apply the distributive law of multiplication and get
$(a - b)(a + b) = {a^2} + ab - ab - {b^2}$
Further, combining the like terms, we get
$(a - b)(a + b) = {a^2} - {b^2}$
Which turns out to be the left hand side of the formulation.

Note:
We can calculate any sort of number involved in these types by just applying the formulation. If the student wants to cross check the calculated answer, then the student can directly square off the numbers and then subtract them. If the answer comes out to be the same, then the calculations are correct, otherwise the student must recheck once and find out for the error he or she had committed.