Question

Cho \(\frac{x}{z}=\frac{z}{y}\)  CMR \(\frac{x^2+z^2}{y^2+z^2}=\frac{x}{y}\)

So sánh\(2^{332}\) và \(3^{223}\)

Answer 1

a/

\(\frac{x}{z}=\frac{z}{y}\Rightarrow\frac{x^2}{z^2}=\frac{z^2}{y^2}=\frac{x^2+z^2}{z^2+y^2}\) (1)

Mà \(\frac{x}{z}=\frac{z}{y}\Rightarrow\frac{x^2}{z^2}=\frac{x}{z}.\frac{z}{y}=\frac{x}{y}\) (2)

Từ 91) và (2) \(\Rightarrow\frac{x^2+z^2}{y^2+z^2}=\frac{x}{y}\left(dpcm\right)\)

Answer 2

\(2^{332}< 2^{333}=\left(2^3\right)^{111}=8^{111}\)

\(3^{223}>3^{222}=\left(3^2\right)^{111}=9^{111}\)

\(\Rightarrow3^{223}>9^{111}>8^{111}>2^{332}\)

Answer 3

a) Từ \(\frac{x}{z}=\frac{z}{y}\)\(\Rightarrow\left(\frac{x}{z}\right)^2=\left(\frac{z}{y}\right)^2\)

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

\(\left(\frac{x}{z}\right)^2=\left(\frac{z}{y}\right)^2=\frac{x^2}{z^2}=\frac{z^2}{y^2}=\frac{x^2+z^2}{y^2+z^2}\)

mà \(\left(\frac{x}{z}\right)^2=\frac{x}{z}.\frac{x}{z}=\frac{x}{z}.\frac{z}{y}=\frac{x}{y}\)

\(\Rightarrow\frac{x^2+z^2}{y^2+z^2}=\frac{x}{y}\)( đpcm )

b) Ta có: \(2^{332}< 2^{333}=2^{3.111}=\left(2^3\right)^{111}=8^{111}\)

              \(3^{223}>3^{222}=3^{2.111}=\left(3^2\right)^{111}=9^{111}\)

Vì \(8< 9\)\(\Rightarrow8^{111}< 9^{111}\)

\(\Rightarrow2^{332}< 8^{111}< 9^{111}< 3^{223}\)

\(\Rightarrow2^{332}< 3^{223}\)