Thu gọn các biểu thức sau: 16-a^2/a^2+8a+16 : a-4/2a+4 * a+4/a+2

Giải SBT Toán 8 Bài 7: Nhân, chia phân thức

Bài 7 trang 25 SBT Toán 8 Tập 1:Thu gọn các biểu thức sau:

a)$\frac{16-a^2}{a^2+8a+16}:\frac{a-4}{2a+4}.\frac{a+4}{a+2}$;

b)$\frac{a^2-ab+b^2}{b^2-a^2}.\frac{a+b}{a^3+b^3}:\frac{a+b}{a-b}$;

c)$\left(\frac{2a}{a-2}-\frac{a}{a+2}\right).\frac{a^2-4}{a}$;

d)$\left(\frac{1}{a^2}-\frac{1}{ab}\right).\frac{a{b}^2}{a-b}$.

Lời giải:

a)$\frac{16-a^2}{a^2+8a+16}:\frac{a-4}{2a+4}.\frac{a+4}{a+2}=\frac{(4+a)(4-a)}{{(a+4)}^2}.\frac{2a+4}{a-4}.\frac{a+4}{a+2}$

$=\frac{-(a+4)(a-4).2(a+2).\left(a+4\right)}{{(a+4)}^2.\left(a-4\right).\left(a+2\right)}=-2$.

b)$\frac{a^2-ab+b^2}{b^2-a^2}.\frac{a+b}{a^3+b^3}:\frac{a+b}{a-b}$

$=-\frac{a^2-ab+b^2}{a^2-b^2}.\frac{a+b}{a^3+b^3}.\frac{a-b}{a+b}$

$=-\frac{a^2-ab+b^2}{\left(a+b\right)\left(a-b\right)}.\frac{a+b}{\left(a+b\right)\left(a^2-ab+b^2\right)}.\frac{a-b}{a+b}$

$=\frac{-1}{{(a+b)}^2}$.

c)$\left(\frac{2a}{a-2}-\frac{a}{a+2}\right).\frac{a^2-4}{a}$

$=\left(\frac{2a}{a-2}-\frac{a}{a+2}\right).\frac{\left(a+2\right)\left(a-2\right)}{a}$

$=\frac{2a}{a-2}.\frac{\left(a+2\right)(a-2)}{a}-\frac{a}{a+2}.\frac{\left(a+2\right)(a-2)}{a}$

$=2(a+2)-(a-2)=2a+4-a+2=a+6$.

d)$\left(\frac{1}{a^2}-\frac{1}{ab}\right).\frac{a{b}^2}{a-b}$

$=\left(\frac{b}{a^{2}b}-\frac{a}{a^{2}b}\right).\frac{a{b}^2}{a-b}$

$=\frac{b-a}{a^{2}b}.\frac{a{b}^2}{a-b}=\frac{-(a-b).ab.b}{ab.a.(a-b)}=-\frac{b}{a}$.