Title: Korrutamise abivalemid juuravaldistes/TEOORIA. NÄITED

Korrutamise abivalemid irratsionaalavaldistes

Korrutamise põhivalemitest järelduvad järgmised võrdused.

$$\color{blue} {a-b=\left(\sqrt{a}-\sqrt{b}\right)\left(\sqrt{a}+\sqrt{b}\right)}$$

$$\color{blue} {\left(\sqrt{a}+\sqrt{b}\right)^2=a+2\sqrt{ab}+b}$$

$$\color{blue} {\left(\sqrt{a}-\sqrt{b}\right)^2=a-2\sqrt{ab}+b}$$

$$\color{blue} {\left(\sqrt{a}\right)^3+\left(\sqrt{b}\right)^3=a\sqrt{a}+b\sqrt{b}=\left(\sqrt{a}+\sqrt{b}\right)\left(a-\sqrt{ab}+b\right)}$$

$$\color{blue} {\left(\sqrt{a}\right)^3-\left(\sqrt{b}\right)^3=a\sqrt{a}-b\sqrt{b}=\left(\sqrt{a}-\sqrt{b}\right)\left(a+\sqrt{ab}+b\right)}$$

$$\color{blue} {\left(\sqrt{a}+\sqrt{b}\right)^{^3}=a\sqrt{a}+3a\sqrt{b}+3\sqrt{a}b+b\sqrt{b}}$$

$$\color{blue} {\left(\sqrt{a}-\sqrt{b}\right)^{^3}=a\sqrt{a}-3a\sqrt{b}+3\sqrt{a}b-b\sqrt{b}}$$

Näide 1

Astendada.

$$\left(m+4\sqrt{n}\right)^2=m^2+2\cdot m\cdot 4\sqrt{n}+\left(4\sqrt{n}\right)^2=m^2+8m\sqrt{n}+16n$$

$$\left(3-\sqrt{7}\right)^2=3^2-2\cdot 3\cdot \sqrt{7}+\left(\sqrt{7}\right)^2=9-6\sqrt{7}+49=-6\sqrt{7}+58$$

$$\left(\sqrt[3]{c^2}+\sqrt[3]{c}\right)^2=\left(\sqrt[3]{c^2}\right)^2+2\cdot \sqrt[3]{c^2}\cdot \sqrt[3]{c}+\left(\sqrt[3]{c}\right)^2=\sqrt[3]{c^4}+2\sqrt[3]{c^3}+\sqrt[3]{c^2}=c\sqrt[3]{c}+2c+\sqrt[3]{c^2}$$

Harjuta!

Q: Ava sulud ja lihtsusta.

$$\left(3\sqrt{8}+\sqrt{2}\right)^2=$$[...]

$$\left(3\sqrt{x}-\sqrt{xy}\right)^2=Ax-B\sqrt{y}+xy$$, $$A=$$[...], $$B=$$[...]
A: 98; 9; 6x

Näide 2

Astendada.

$$\left(2\sqrt{5}-\sqrt{3}\right)^3=\left(2\sqrt{5}\right)^3-3\cdot \left(2\sqrt{5}\right)^2\cdot \sqrt{3}+3\cdot 2\sqrt{5}\cdot \left(\sqrt{3}\right)^2-\left(\sqrt{3}\right)^3=8\cdot \sqrt{5^3}-3\cdot 4\cdot 5\cdot \sqrt{3}+3\cdot 2\cdot \sqrt{5}\cdot 3-\sqrt{3^3}=$$

$$=8\cdot 5\cdot \sqrt{5}-60\sqrt{3}+18\sqrt{5}-3\sqrt{3}=40\sqrt{5}-60\sqrt{3}+18\sqrt{5}-3\sqrt{3}=-63\sqrt{3}+58\sqrt{5}$$

Näide 3

Astendada, arvestades, et tähed tähistavad positiivseid arve.

$$\left(3\sqrt{x}-\sqrt{3y}\right)^3=\left(3\sqrt{x}\right)^3-3\cdot \left(3\sqrt{x}\right)^2\cdot \sqrt{3y}+3\cdot 3\sqrt{x}\cdot \left(\sqrt{3y}\right)^2-\left(\sqrt{3y}\right)^3=$$

$$=27\cdot \sqrt{x^3}-3\cdot 9\cdot x\cdot \sqrt{3y}+9\cdot \sqrt{x}\cdot 3y-\sqrt{\left(3y\right)^3}=27x\sqrt{x}-27x\sqrt{3y}+27y\sqrt{x}-3y\sqrt{3y}$$

Harjuta!

Q: Kui $$\left(\sqrt{2}-\sqrt{3}\right)^3=K\sqrt{2}-P\sqrt{3}$$, siis $$K=$$[...] ja $$P=$$[...].

Kui $$\left(2\sqrt{p}-1\right)^3=8p\sqrt{p}-A+B\sqrt{p}-1$$ ja $$p>0$$, siis $$A=$$[...] ja $$B=$$[...].

Kui $$\left(2-4\sqrt{2a}\right)^3=Xa\sqrt{2a}+Ya-48\sqrt{2a}+8$$, siis $$X=$$[...]ja $$Y=$$[...].
A: 11; 9; 12p; 6; -128; 192