Simplify: (i)(a²−b²)² (ii)(2x+5)²−(2x−5)² (iii)(7m−8n)²+(7m+8n)² (iv)(4m+5n)²+(4n+5m)² (v)(2.5p−1.5q)²−(1.5p−2.5q)² (vi)(ab+bc)²−2ab²c (vii)(m²−n²/m)²+2m³n²

i)(a²−b²)²=(a²)²+(b²)²−2(a²)(b²) ∝[(a+b)²=a²+b²+2ab]=a⁴+b⁴−2a²b²
ii)(2x+5)²−(2x−5)²
Let A=2x+5 and B=2x−5 ∴(2x+5)²−(2x−5)²=A²−B²=(A+B)(A−B) [∝a²−b²=(a+b)(a−b)]=[2x+5+(2x−5)][2x+5−(2x−5)]=[2x+5+2x−5][2x+5−2x+5]=(4x)(10)=40x
iii)(7m−8n)²+(7m+8n)²=(7m)²+(8n)²−2(7m)(8n)+(7m)²+(8n)²+2(7m)(8n)=49m²+64n²−112mn+49m²+64n²+112mn=98m²+128n²
iv)(4m+5n)²+(5m+4n)²=(4m)²+(5n)²+2(4m)(5n)+(5m)²+(4n)²+2(5m)(4n) [∝(a−b)²=a²−2ab+b²]=16m²+25n²+40mn+25m²+16n²+40mn=41m²+41n²+80mn
v)(2.5p−1.5q)²−(1.5p−2.5q)²=(2.5p)²+(1.5q)²−2(2.5p)(1.5q)−[(1.5p)²+(2.5q)²−2(1.5p)(2.5q)]=6.25p²+2.25q²−7.5pq−2.25p²−6.25q²+7.5pq=4p²−4q²=4(p²−q²)=4(p−q)(p+q)
vi)(ab+bc)²−2ab²c=(ab)²+(bc)²+2(ab)(bc)−2ab²c=a²b²+b²c²+2ab²c−2ab²c=b²(a²+c²)
vii)(m²−n²/m)²+2m³n²=(m²)²+(n²/m)²−2(m²)(n²/m)+2m³n²=m⁴+n⁴/m²−2m³n²/m+2m³n²=m⁴+n⁴/m²−2mn²+2m³n²=m²(m²-n⁴)