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Simplify:
(i) $(x - y) (x + y) (x^2 + y^2) (x^4 + y^4)$
(ii) $(2x - 1) (2x + 1) (4x^2 + 1) (16x^4 + 1)$
(iii) $(7m - 8n)^2 + (7m + 8n)^2$
(iv) $(2.5p - 1.5q)^2 - (1.5p - 2.5q)^2$
(v) $(m^2 - n^2m)^2 + 2m^3n^2$
Given:
(i) $(x - y) (x + y) (x^2 + y^2) (x^4 + y^4)$
(ii) $(2x - 1) (2x + 1) (4x^2 + 1) (16x^4 + 1)$
(iii) $(7m - 8n)^2 + (7m + 8n)^2$
(iv) $(2.5p - 1.5q)^2 - (1.5p - 2.5q)^2$
(v) $(m^2 - n^2m)^2 + 2m^3n^2$
To do:
We have to simplify the given expressions.
Solution:
Here, we have to simplify the given expressions. By using the algebraic identities $(a+b)^2=a^2+2ab+b^2$, $(a-b)^2=a^2-2ab+b^2$ and $(a+b)(a-b)=a^2-b^2$, we can reduce the given expressions and simplify them.
$(a+b)^2=a^2+2ab+b^2$.............(I)
$(a-b)^2=a^2-2ab+b^2$.............(II)
$(a+b)(a-b)=a^2-b^2$................(III)
(i) The given expression is $(x - y) (x + y) (x^2 + y^2) (x^4 + y^4)$.
$(x - y) (x + y) (x^2 + y^2) (x^4 + y^4)=(x^2-y^2)(x^2+y^2)(x^4+y^4)$ [Using (III)]
$(x - y) (x + y) (x^2 + y^2) (x^4 + y^4)=[(x^2)^2-(y^2)^2](x^4+y^4)$ [Using (III)]
$(x - y) (x + y) (x^2 + y^2) (x^4 + y^4)=(x^4-y^4)(x^4+y^4)$
$(x - y) (x + y) (x^2 + y^2) (x^4 + y^4)=(x^4)^2-(y^4)^2$ [Using (III)]
$(x - y) (x + y) (x^2 + y^2) (x^4 + y^4)=x^8-y^8$
(ii) The given expression is $(2x - 1) (2x + 1) (4x^2 + 1) (16x^4 + 1)$.
$(2x - 1) (2x + 1) (4x^2 + 1) (16x^4 + 1)=[(2x)^2 - (1)^2] (4x^2 + 1) (16x^4 + 1)$ [Using (III)]
$(2x - 1) (2x + 1) (4x^2 + 1) (16x^4 + 1)=(4x^2 - 1) (4x^2 + 1) (16x^4 + 1)$
$(2x - 1) (2x + 1) (4x^2 + 1) (16x^4 + 1)=[(4x^2)^2 - (1)^2] (16x^4 + 1)$ [Using (III)]
$(2x - 1) (2x + 1) (4x^2 + 1) (16x^4 + 1)=(16x^4 - 1) (16x^4 + 1)$
$(2x - 1) (2x + 1) (4x^2 + 1) (16x^4 + 1)=(16x^4)^2 - (1)^2$ [Using (III)]
$(2x - 1) (2x + 1) (4x^2 + 1) (16x^4 + 1)=256x^8 - 1)$
(iii) The given expression is $(7m - 8n)^2 + (7m + 8n)^2$.
$(7m - 8n)^2 + (7m + 8n)^2=[(7m)^2-2(7m)(8n)+(8n)^2]+[(7m)^2+2(7m)(8n)+(8n)^2]$ [Using (I) and (II)]
$(7m-8n)^2+(7m+8n)^2=(49m^2-112mn+64n^2)+(49m^2+112mn+64n^2)$
$(7m-8n)^2+(7m+8n)^2=49m^2+49m^2-112mn+112mn+64n^2+64n^2$
$(7m-8n)^2+(7m+8n)^2=98m^2+128n^2$
(iv) The given expression is $(2.5p - 1.5q)^2 - (1.5p - 2.5q)^2$.
$(2.5p - 1.5q)^2 - (1.5p - 2.5q)^2=[(2.5p - 1.5q)+(1.5p - 2.5q)][(2.5p - 1.5q)-(1.5p - 2.5q)]$ [Using (III)]
$(2.5p - 1.5q)^2 - (1.5p - 2.5q)^2=(2.5p+1.5p-1.5q-2.5q)(2.5p-1.5p- 1.5q+ 2.5q)$
$(2.5p - 1.5q)^2 - (1.5p - 2.5q)^2=(4p-4q)(p+ q)$
$(2.5p - 1.5q)^2 - (1.5p - 2.5q)^2=4(p-q)(p+ q)$ (Taking $4$ common)
$(2.5p - 1.5q)^2 - (1.5p - 2.5q)^2=4(p^2-q^2)$ [Using (III)]
(v) The given expression is $(m^2 - n^2m)^2 + 2m^3n^2$.
$(m^2-n^2m)^2+2m^3n^2=[(m^2)^2-2(m^2)(n^2m)+(n^2m)^2]+2m^3n^2$ [Using (II)]
$(m^2-n^2m)^2+2m^3n^2=m^4-2m^3n^2+n^4m^2+2m^3n^2$
$(m^2-n^2m)^2+2m^3n^2=m^4-2m^3n^2+2m^3n^2+n^4m^2$
$(m^2-n^2m)^2+2m^3n^2=m^4+n^4m^2$
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