2 zadania z liczb rzeczywistych.

[Wikers2]

1. Oblicz:
a) \((3 \sqrt{2} - 4)(3 \sqrt{2} +4)\)
b) \((3 + 2 \sqrt{5} )(3 - 2 \sqrt{5})\)
c) \((2+ \sqrt{3}) ^{2} (7- 4\sqrt{3} )\)
d) \((1-2 \sqrt{2})^2(9+4 \sqrt{2})\)
e) \((3+ \sqrt{7})^2 - (3- \sqrt{7})^2\)
f) \((2 \sqrt{2} - \sqrt{3} )^2 + (2 \sqrt{3} + \sqrt{2})^2\)
2. Oblicz, jaki procent liczby x stanowi liczba y.
a) x = \(( \frac{2}{3} - \frac{5}{12}) : ( \frac{1}{4} - \frac{3}{16})\) , y = \(\frac{ \sqrt{3} * \sqrt[3]{9} }{ \sqrt[6]{3} }\)
b) x = \(3 ^{-1} * 5 ^{ -\frac{1}{3} } * \frac{81^ \frac{1}{4} }{ \sqrt[3]{25} }\), y = \(( \frac{2}{3} ) ^{-3} * ( \frac{ \sqrt[4]{4} }{3} )^2\)

Dziękuje za pomoc

[eresh]

1. Oblicz:
a) \((3 \sqrt{2} - 4)(3 \sqrt{2} +4)\)

\((3 \sqrt{2} - 4)(3 \sqrt{2} +4)=(3\sqrt{2})^2-4^2=18-16=2\)

[eresh]

1. Oblicz:

b) \((3 + 2 \sqrt{5} )(3 - 2 \sqrt{5})\)

\((3+2\sqrt{5})(3-2\sqrt{5})=3^2-(2\sqrt{5})^2=9-20=-11\)

[eresh]

1. Oblicz:
c) \((2+ \sqrt{3}) ^{2} (7- 4\sqrt{3} )\)

\((2+ \sqrt{3}) ^{2} (7- 4\sqrt{3} )=(4+4\sqrt{3}+3)(7-4\sqrt{3})=(7+4\sqrt{3})(7-4\sqrt{3})=49-48=1\)

[eresh]

1. Oblicz:
d) \((1-2 \sqrt{2})^2(9+4 \sqrt{2})\)

\((1-2 \sqrt{2})^2(9+4 \sqrt{2})=(1-4\sqrt{2}+8)(9+4\sqrt{2})=(9-4\sqrt{2})(9+4\sqrt{2})=81-32=49\)

[eresh]

1. Oblicz:
e) \((3+ \sqrt{7})^2 - (3- \sqrt{7})^2\)

\((3+\sqrt{7})^2-(3-\sqrt{7})^2=(3+\sqrt{7}-3+\sqrt{7})(3+\sqrt{7}+3-\sqrt{7})=2\sqrt{7}\cdot 6=12\sqrt{7}\)

[eresh]

1. Oblicz:
f) \((2 \sqrt{2} - \sqrt{3} )^2 + (2 \sqrt{3} + \sqrt{2})^2\)

\((2 \sqrt{2} - \sqrt{3} )^2 + (2 \sqrt{3} + \sqrt{2})^2=8-4\sqrt{6}+3+12+4\sqrt{6}+2=25\)

[eresh]

2. Oblicz, jaki procent liczby x stanowi liczba y.
a) x = \(( \frac{2}{3} - \frac{5}{12}) : ( \frac{1}{4} - \frac{3}{16})\) , y = \(\frac{ \sqrt{3} * \sqrt[3]{9} }{ \sqrt[6]{3} }\)

\(x=( \frac{2}{3} - \frac{5}{12}) : ( \frac{1}{4} - \frac{3}{16})=\frac{8-5}{12}:\frac{4-3}{16}=\frac{3}{12}\cdot\frac{16}{1}=4\)

\(y=\frac{3^{0,5}\cdot 3^{\frac{2}{3}}}{3^{\frac{1}{6}}}=\frac{3^{\frac{7}{6}}}{3^{\frac{1}{6}}}=3\)

\(\frac{y}{x}\cdot 100\%=\frac{3}{4}\cdot 100\%=75\%\)

[eresh]

2. Oblicz, jaki procent liczby x stanowi liczba y.
b) x = \(3 ^{-1} * 5 ^{ -\frac{1}{3} } * \frac{81^ \frac{1}{4} }{ \sqrt[3]{25} }\), y = \(( \frac{2}{3} ) ^{-3} * ( \frac{ \sqrt[4]{4} }{3} )^2\)

\(x=\frac{3^{-1}\cdot 5^{-\frac{1}{3}}\cdot 3}{5^{\frac{2}{3}}}=5^{-1}=\frac{1}{5}\)

\(y=(\frac{2}{3})^{-3}\cdot (\frac{2^{0,5}}{3})^2=\frac{3^3}{2^3}\cdot\frac{2}{3^2}=\frac{3}{4}\)

\(\frac{y}{x}\cdot 100\%=\frac{0,75}{0,2}\cdot 100\%=375\%\)