Question

simplify (a^2-b^2)^2

Answer 1

Step 1 :

b2
Simplify ——
2
Equation at the end of step 1 :

b2
((a2) - (—— • a)) - 2b
2
Step 2 :

Rewriting the whole as an Equivalent Fraction :

2.1 Subtracting a fraction from a whole

Rewrite the whole as a fraction using 2 as the denominator :

a2 a2 • 2
a2 = —— = ——————
1 2
Equivalent fraction : The fraction thus generated looks different but has the same value as the whole

Common denominator : The equivalent fraction and the other fraction involved in the calculation share the same denominator

Adding fractions that have a common denominator :

2.2 Adding up the two equivalent fractions
Add the two equivalent fractions which now have a common denominator

Combine the numerators together, put the sum or difference over the common denominator then reduce to lowest terms if possible:

a2 • 2 - (ab2) 2a2 - ab2
—————————————— = —————————
2 2
Equation at the end of step 2 :

(2a2 - ab2)
——————————— - 2b
2
Step 3 :

Rewriting the whole as an Equivalent Fraction :

3.1 Subtracting a whole from a fraction

Rewrite the whole as a fraction using 2 as the denominator :

2b 2b • 2
2b = —— = ——————
1 2
Step 4 :

Pulling out like terms :

4.1 Pull out like factors :

2a2 - ab2 = a • (2a - b2)

Trying to factor as a Difference of Squares :

4.2 Factoring: 2a - b2

Theory : A difference of two perfect squares, A2 - B2 can be factored into (A+B) • (A-B)

Proof : (A+B) • (A-B) =
A2 - AB + BA - B2 =
A2 - AB + AB - B2 =
A2 - B2

Note : AB = BA is the commutative property of multiplication.

Note : - AB + AB equals zero and is therefore eliminated from the expression.

Check : 2 is not a square !!

Ruling : Binomial can not be factored as the
difference of two perfect squares

Adding fractions that have a common denominator :

4.3 Adding up the two equivalent fractions

a • (2a-b2) - (2b • 2) 2a2 - ab2 - 4b
—————————————————————— = ——————————————
2 2
Trying to factor a multi variable polynomial :

4.4 Factoring 2a2 - ab2 - 4b

Try to factor this multi-variable trinomial using trial and error

Factorization fails

Final result :

2a2 - ab2 - 4b
——————————————
2

Answer 2

Answer:

a^4 - 2a^2b^2 + (b^2)^2

Step-by-step explanation:

using the property -

{ (a-b)^2 = a^2 - 2ab + b^2 }

= (a^2)^2 - 2(a^2) (b^2) + (b^2)^2

= a^4 - 2a^2 b^2 + b^4

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