Question

Simplify ${({a^2} - {b^2})^2}$
A) ${a^4} - 2{a^2}{b^2} + {b^4}$
B) ${a^4} + 2{a^2}{b^2} + {b^4}$
C) ${a^4} - 2{a^2}{b^2} - {b^4}$
D) ${a^4} + 2{a^2}{b^2} - {b^4}$

Answer 1

Hint: According to given in the question we have to simplify or find the value of the given expression ${({a^2} - {b^2})^2}$so, to simplify the given expression first of all we have to use the formula as given below:
Formula used:
${(x - y)^2} = {x^2} + {y^2} - 2xy$ ..............(1)
We can obtain the formula given above by the simple calculations as given below:
$
   \Rightarrow {(x - y)^2} = (x - y) \times (x - y) \\
   \Rightarrow {(x - y)^2} = x(x - y) - y(x - y) \\
   \Rightarrow {(x - y)^2} = {x^2} - xy - xy + {y^2} \\
   \Rightarrow {(x - y)^2} = {x^2} + {y^2} - 2xy \\
 $
Hence, with the help of the formula (1) given above we can simplify the value of ${({a^2} - {b^2})^2}$.

Complete step by step answer:
Step 1: To simplify the given expression ${({a^2} - {b^2})^2}$first of all we have to use the formula (1) as mentioned in the solution hint.
Step 2: From the formula (1) let’s imagine a is x and b is y now use the formula,
$ \Rightarrow {({a^2} - {b^2})^2} = {({a^2})^2} + {({b^2})^2} - 2{a^2}{b^2}$ …………………(2)
Step 3: On solving the obtained expression (2),
$ \Rightarrow {({a^2} - {b^2})^2} = {a^4} + {b^4} - 2{a^2}{b^2}$

Hence, with the help of the formula (1) we have obtained the value or we have simplified the given expression $ \Rightarrow {({a^2} - {b^2})^2} = {a^4} + {b^4} - 2{a^2}{b^2}$

Note:
Alternative method:
We can also simplify the given expression by the calculations as given below:
Step 1: As given in the question to simplify the expression ${({a^2} - {b^2})^2}$ first of all we have to open the square.
$ \Rightarrow {({a^2} - {b^2})^2} = ({a^2} - {b^2}) \times ({a^2} - {b^2})$
Step 2: Now, we have to multiply each term of the obtained expression in step 1.
$
   \Rightarrow {({a^2} - {b^2})^2} = {a^2}({a^2} - {b^2}) - {b^2}({a^2} - {b^2}) \\
   \Rightarrow {({a^2} - {b^2})^2} = {a^2} \times {a^2} - {a^2} \times {b^2} - {b^2} \times {a^2} + {b^2} \times {b^2} \\
   \Rightarrow {({a^2} - {b^2})^2} = {a^4} + {b^4} - 2{a^2}{b^2} \\
 $
Step 3: Hence, from the step 2 we have simplified the given expression:
$ \Rightarrow {({a^2} - {b^2})^2} = {a^4} + {b^4} - 2{a^2}{b^2}$