Question

[Nguyễn Trân]

1)

a) So sánh (-99)²⁰ và 9999¹⁰⁰

b) Chứng tỏ A = 27²⁰ + 3⁶¹ + 9³¹ chia hết cho 13

2)

Cho\(\dfrac{a}{b}\)= \(\dfrac{b}{c}\)=\(\dfrac{c}{2011}\) =\(\dfrac{2011}{a}\) và a+b+c khác 0. Tính a+b-c

3) Tính M = \(\dfrac{4}{237}-\dfrac{4}{2371}+\dfrac{4}{23711}\) phần \(\dfrac{-5}{237}+\dfrac{5}{2371}-\dfrac{5}{23711}\)

Answer 1

[Mới vô]

3,

\(M=\dfrac{\dfrac{4}{237}-\dfrac{4}{2371}+\dfrac{4}{23711}}{\dfrac{-5}{237}+\dfrac{5}{2371}-\dfrac{5}{23711}}=\dfrac{\left(-4\right)\cdot\left(\dfrac{-1}{237}+\dfrac{1}{2371}-\dfrac{1}{23711}\right)}{5\cdot\left(\dfrac{-1}{237}+\dfrac{1}{2371}-\dfrac{1}{23711}\right)}=\dfrac{-4}{5}\)

Vậy \(M=\dfrac{-4}{5}\)

2,

Áp dụng tính chất dãy tỉ số bằng nhau ta có:

\(\dfrac{a}{b}=\dfrac{b}{c}=\dfrac{c}{2011}=\dfrac{2011}{a}=\dfrac{a+b+c+2011}{b+c+2011+a}=\dfrac{a+b+c+2011}{a+b+c+2011}=1\)

\(\dfrac{a}{b}=1\Rightarrow a=b\left(1\right)\\ \dfrac{b}{c}=1\Rightarrow b=c\left(2\right)\)

Từ (1) và (2) ta có: \(a=c\)

\(\Rightarrow a+b-c=a+a-a=a\)

1)

b)

\(A=27^{20}+3^{61}+9^{31}\\ =\left(3^3\right)^{20}+3^{61}+\left(3^2\right)^{31}\\ =3^{60}+3^{61}+3^{62}\\ =3^{60}\cdot\left(1+3+3^2\right)\\ =3^{60}\cdot\left(1+3+9\right)\\ =3^{60}\cdot13⋮13\)

Vậy \(A⋮13\)

a,

\(\left(-99\right)^{20}=\left(-99\right)^{2\cdot10}=\left[\left(-99\right)^2\right]^{10}=9801^{10}\\ 9999^{100}=\left(9999^{10}\right)^{10}>\left(9999^{10}\right)^1=9999^{10}\)

Vì \(9801^{10}< 9999^{10}< \left(9999^{10}\right)^{10}=9999^{100}\Rightarrow\left(-99\right)^{20}< 9999^{100}\)

Vậy \(\left(-99\right)^{20}< 9999^{100}\)

Answer 2

[Huy Thắng Nguyễn]

1/

a) (-99)²⁰ = 99²⁰

Vì 99 < 9999

20 < 100

Nên 99²⁰ < 9999¹⁰⁰

Vậy (-99)²⁰ < 9999¹⁰⁰

b) \(A=27^{20}+3^{61}+9^{31}\)

\(=\left(3^3\right)^{20}+3^{61}+\left(3^2\right)^{31}\)

\(=3^{60}+3^{61}+3^{62}\)

\(=3^{60}\left(1+3+3^2\right)\)

\(=3^{60}.13⋮13\)

Vậy A chia hết cho 13.

2) Áp dụng tính chất dãy tỉ số bằng nhau ta có:

\(\dfrac{a}{b}=\dfrac{b}{c}=\dfrac{c}{2011}=\dfrac{2011}{a}=\dfrac{a+b+c+2011}{b+c+2011+a}=1\)

\(\Rightarrow\dfrac{a}{b}=1;\dfrac{b}{c}=1\Rightarrow a=b=c\) (*)

Thay (*) vào a + b - c: a + a - a = a

Vậy a + b - c = a.

3. \(M=\dfrac{\dfrac{4}{237}-\dfrac{4}{2371}+\dfrac{4}{23711}}{-\dfrac{5}{237}+\dfrac{5}{2371}-\dfrac{5}{23711}}\)

\(=\dfrac{4\left(\dfrac{1}{237}-\dfrac{1}{2371}+\dfrac{1}{23711}\right)}{-5\left(\dfrac{1}{237}-\dfrac{1}{2371}+\dfrac{1}{23711}\right)}\)

\(=-\dfrac{4}{5}\)