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4.Simplify: (i)$$(a^2 – b^2)^2$$ (ii) $$(2x + 5)^2 – (2x – 5)^2$$ (iii) $$(7m – 8n)^2 + (7m + 8n)^2$$ (iv) $$(4m + 5n)^2 + (5m + 4n)^2$$ (v) $$(2.5p – 1.5q)^2 – (1.5p – 2.5q)^2$$ (vi) $$(ab + bc)^2 – 2ab^2c$$ (vii) $$(m^2 – n^2m)^2 + 2m^3n^2$$

(i)Given:$$(a^2-b^2)^2$$
$$=(a^2)^2-2a^2b^2+(b^2)^2\quad$$[use $$(a-b)^2=a^2-2ab+b^2$$]

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

(ii)Given:$$(2x + 5)^2 – (2x – 5)^2$$

$$=[(2x)^2+2(2x)(5)+(5)^2]-[(2x)^2-2(2x)(5)+(5)^2]$$

$$=[4x^2+20x+25]-[4x^2-20x+25]$$

$$=4x^2+20x+25-4x^2+20x-25]$$

$$=(4x^2-4x^2)+(+20x+20x)+(+25-25)\quad$$[Placing same terms together]

$$=40x$$

(iii)Given:$$(7m – 8n)^2 + (7m + 8n)^2$$

$$=[(7m)^2-2(7m)(8n)+(8n)^2] + [(7m)^2+2(7m)(8n)+(8n)^2]$$

$$=[49m^2-112mn+64n^2]+[49m^2+112mn+64n^2]$$

$$=(49m^2+49m^2)+(-112mn+112mn)+(64n^2+64n^2)\quad$$[Placing same terms together]

$$=98m^2+128n^2$$

(iv)Given:$$(4m + 5n)^2 + (5m + 4n)^2$$

$$=[(4m)^2+2(4m)(5n)+(5n)^2] + [(5m)^2+2(5m)(4n)+(4n)^2]$$

$$=[16m^2+40mn+25n^2]+[25m^2+40mn+16n^2]$$

$$=(16m^2+25m^2)+(40mn+40mn)+(25n^2+16n^2)\quad$$[Placing same terms together]

$$=41m^2+80mn+41n^2$$

(v)Given: $$(2.5p – 1.5q)^2 – (1.5p – 2.5q)^2$$

$$=[(2.5p)^2-2(2.5p)(1.5q)+(1.5q)^2] - [(1.5p)^2-2(1.5p)(2.5q)+(2.5q)^2]$$

$$=[6.25p^2-7.5pq+2.25q^2]-[2.25p^2-7.5pq+6.25q^2]$$

$$=6.25p^2-7.5pq+2.25q^2-2.25p^2+7.5pq-6.25q^2$$

$$=(6.25p^2-2.25p^2)+(-7.5pq+7.5pq)+(2.25q^2-6.25q^2)\quad$$[Placing same terms together]

$$=4p^2-4q^2$$

(vi)Given:$$(ab + bc)^2 – 2ab^2c$$

$$=(ab)^2+2(ab)(bc)+(bc)^2-2ab^2c$$

$$=a^2b^2+2ab^2c+b^2c^2-2ab^2c$$

$$=a^2b^2+b^2c^2$$

(vii)Given:$$(m^2 – n^2m)^2 + 2m^3n^2$$
$$=(m^2)^2-2(m^2)(n^2m)+(n^2m)^2+2m^3n^2$$

$$=m^4-2m^3n^2+n^4m^2+2m^3n^2$$

$$=m^4+n^4m^2$$