Задание 1.15

Question

Упростите выражение:
$$1) \left(a^{\sqrt{5}}+2\right)\left(a^{\sqrt{5}}-2\right)-\left(a^{\sqrt{5}}+3\right)^{2}$$
$$2) \frac{a^{2 \sqrt{7}}-a^{\sqrt{7}}}{a^{4 \sqrt{7}}-a^{3 \sqrt{7}}}$$
$$3) \frac{a^{2 \sqrt{3}}-b^{2 \sqrt{2}}}{\left(a^{\sqrt{3}}+b^{\sqrt{2}}\right)^{2}}+1$$
$$4) \frac{a^{\sqrt[3]{24}}-1}{a^{\sqrt[3]{3}}-1}-\frac{a^{\sqrt[3]{81}}+1}{a^{\sqrt[3]{3}}+1}$$.

Gimli · Accepted Answer

$$1)\left(a^{\sqrt{5}}+2\right)\left(a^{\sqrt{5}}-2\right)-\left(a^{\sqrt{5}}+3\right)^{2}=a^{2 \sqrt{5}}-4-\left(a^{2\sqrt{5}}+6 a^{\sqrt{5}}+9\right)=$$
 $$=a^{2 \sqrt{5}}-4-a^{2 \sqrt{5}}-6 a^{\sqrt{5}}-9=-6 a^{\sqrt{5}}-13$$
$$2) \frac{a^{2 \sqrt{7}}-a^{\sqrt{7}}}{a^{4 \sqrt{7}}-a^{3\sqrt{ 7}}}=\frac{a^{\sqrt{7}}\left(a^{2}-a\right)}{a^{\sqrt{7}}\left(a^{4}-a^{3}\right)}=\frac{a^{2}-a}{a^{4}-a^{3}}=\frac{a(a-1)}{a^{3}(a-1)}=\frac{1}{a^{2}}$$
$$3) \frac{a^{2 \sqrt{3}}-b^{2 \sqrt{2}}}{\left(a^{\sqrt{3}}+b^{\sqrt{2}}\right)^{2}}+1=$$
 $$=\frac{a^{2 \sqrt{3}}-b^{2 \sqrt{2}}}{\left(a^{\sqrt{3}}+b^{\sqrt{2}}\right)^{2}}+\frac{\left(a^{\sqrt{3}}+b^{\sqrt{2}}\right)^{2}}{\left(a^{\sqrt{3}}+b^{\sqrt{2}}\right)^{2}}=$$
$$=\frac{2 a^{2 \sqrt{3}}+2 a^{\sqrt{3}} b^{\sqrt{2}}}{\left(a^{\sqrt{3}}+b^{\sqrt{2}}\right)^{2}}=\frac{2 a^{\sqrt{3}}\left(a^{\sqrt{3}}+b^{\sqrt{2}}\right)}{\left(a^{\sqrt{3}}+b^{\sqrt{2}}\right)^{2}}$$
$$=\frac{2 a^{\sqrt{3}}}{a^{\sqrt{3}}+b^{\sqrt{2}}}$$
$$4) \frac{a^{\sqrt[3]{24}}-1}{a^{\sqrt[3]{3}}-1}-\frac{a^{\sqrt[3]{81}}+1}{a^{\sqrt[3]{3}}+1}=$$.
$$=\frac{a^{2 \sqrt[3]{3}}-1}{a^{\sqrt[3]{3}}-1}-\frac{a^{3 \sqrt[3]{3}}+1}{a^{ \sqrt[3]{3}}+1}= $$
$$=\frac{\left(a^{\sqrt[3]{3}}\right)^{2}-1^{2}}{a^{ \sqrt[3]{3}}-1}-\frac{\left(a^{\sqrt[3]{3}}\right)^{3}+1^{3}}{a^{\sqrt[3]{3}}+1}= $$
$$=\frac{\left(a^{\sqrt[3]{3}}-1\right)\left(a^{\sqrt[3]{3}}+1\right)}{a^{ \sqrt[3]{3}}-1}-\frac{\left(a^{ \sqrt[3]{3}}+1\right)\left(a^{2 \sqrt[3]{3}}-a^{\sqrt[3]{3}}+1\right)}{a^{\sqrt[3]{3}}+1} $$
$$\Rightarrow a^{\sqrt[3]{3}}+1-\left(a^{2 \sqrt{3}}-a^{\sqrt[3]{3}}+1\right)=$$
$$=2 a^{\sqrt[3]{3}}-a^{2 \sqrt[3]{3}}$$

Задание 1.15

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Задание 1.15

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