Konzultace 17. 12. 2020
Příklad 1: \( \frac {5} {3- (2 \sqrt{2})} = a \times (b \sqrt{c}) \)
Vzorec, ke kterému se snažíme příklad směřovat: \( a^2 - b^2 = (a - b) \times (a + b) \)
\( \frac {5} {3- (2 \sqrt{2})} = \frac {5} {3- (2 \sqrt{2})} \times \frac {3+(2\sqrt{2})}{3+(2\sqrt{2})} \)
Vzorec: \( \frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d} \)
\( \frac {5 \times (3+(2\sqrt{2}))} {(3- (2 \sqrt{2})) \times (3+(2\sqrt{2}))} = \)
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\( a = 3; b = 2 \sqrt{2} \)
\( a^2 = 9; b^2 = 4 \times 2 = 8 \)
\( a^2 - b^2 = 1 \)
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\( \frac {5 \times (3+(2\sqrt{2}))} {1} = 5 \times (3+(2\sqrt{2})) = 5 \times 3 + 5 \times 2 \sqrt{2} = 15 + 10 \sqrt{2} \)
\( a \times (b \sqrt{c}) \)
\( a = 15 \)
\( b = 10 \)
\( c = 2 \)
Příklad 2: \( \frac{\frac{((-x^2)^3)^2 \times (-2)^7} {x^3 \times y^2}} {2^6} = a \times x^b \times y^c \)
Vzorec: \( \frac{\frac{a}{b}} {\frac{c}{d}} = \frac{a}{b} \times \frac{d}{c} \)
\( \frac{\frac{((-x^2)^3)^2 \times (-2)^7} {x^3 \times y^2}} {\frac{2^6}{1}} = \frac{((-x^2)^3)^2 \times (-2)^7} {x^3 \times y^2} \times \frac{1}{2^6} = \)
Vzorec: \( \frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d} \)
\( \frac{((-x^2)^3)^2 \times (-2)^7 \times 1} {x^3 \times y^2 \times 2^6} = \)
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Vzorec: \( (a^r)^s = a^{r \times s} \)
\(((-x^2)^3)^2 = (-x)^{2 \times 3 \times 2} = (-x)^{12} = x^{12}\)
\( -3 * -3 = 9 \)
\( -3 * -3 * -3 = -27 \)
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\( \frac{x^{12} \times (-2)^7 \times 1} {x^3 \times y^2 \times 2^6} = \)
\( \frac{x^{12} \times (-2)^7 } {x^3 \times y^2 \times 2^6} = \)
Vzorec: \( (a^r)^s = a^{r \times s} \)
Vzorec: \( \frac{(a^r)}{(a^s)} = a^{r-s} \)
\( \frac{x^{9} \times y^{-2} \times (-2)^7 } {2^6} = \)
Důležitá poznámka: Využívám toho, že \( (-2)^{jakékoliv sudé číslo} =2^{jakékoliv sudé číslo} \). To ale neplatí pro lichá čísla!
\( \frac{x^{9} \times y^{-2} \times (-2)^7 } {(-2)^6} = x^{9} \times y^{-2} \times (-2)^1 \)
\( x^{9} \times y^{-2} \times (-2)^1 \)
\( (-2) \times x^{9} \times y^{-2} \)
\( a \times x^b \times y^c \)
\( a = -2 \)
\( b = 9 \)
\( c = -2 \)