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رادیکال

قانون اول رادیکال

آخرین ویرایش: 02 تیر 1403

دسته‌بندی: رادیکال

امتیاز:

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مقدمه

توان‌ های كسری تعمیمی از توان های صحیح اند و قواعد محاسبه آنها نباید طوری باشد كه با دانسته های قبلی ما از قواعد توان، در تضاد باشد.

به عبارت دیگر تمام قواعدی كه در محاسبه با توان‌ های صحیح گفته شد، برای محاسبه با توان‌های كسری نیز برقرار است.

$\begin{array}{l} 2^{3} \times 5^{3} = {(2 \times 5)}^{3} = 10^{3} \end{array}$

$2^{\frac{1}{3}} \times 5^{\frac{1}{3}} = {(2 \times 5)}^{\frac{1}{3}} = 10^{\frac{1}{3}} = \sqrt[3]{10}$

تعریف

قضیه

حاصل‌ضرب دو رادیکال با فرجه برابر

اگر $a$ و $b$ دو عدد حقیقی ، $\sqrt[n]{a}$ و $\sqrt[n]{b}$ قابل تعریف باشند، آنگاه:

$\sqrt[n]{a} . \sqrt[n]{b} = \sqrt[n]{a b}$

این قضیه بیان می‌كند كه رادیكال های هم فرجه را می‌توان درهم ضرب كرد.

اثبات

$\sqrt[n]{a^{1}} . \sqrt[n]{b^{1}} = a^{\frac{1}{n}} . b^{\frac{1}{n}} = {(a . b)}^{\frac{1}{n}} = \sqrt[n]{{(a b)}^{1}} = \sqrt[n]{a b}$

به‌عنوان نمونه، جدول زیر را کامل می‌کنیم:

همان‌طور که مشاهده می‌کنید، بر طبق قضیه فوق و جدول بالا جواب ‌های هر دو سطر برابرند.

با انتخاب چند عدد دیگر برای $a$ و $b$ داریم:

تمرین

حاصل ضرب عبارات راديكالی زير را محاسبه کنید.

$\begin{array}{l} \sqrt{28} \times \sqrt{7} \end{array}$

$\begin{array}{l} = \sqrt{28 \times 7} \\ = \sqrt{196} \\ = \sqrt{{(14)}^{2}} \\ = 14 \end{array}$

$\begin{array}{l} \sqrt[3]{36} \times \sqrt[3]{6} \end{array}$

$\begin{array}{l} = \sqrt[3]{36 \times 6} \\ = \sqrt[3]{216} \\ = \sqrt[3]{{(6)}^{3}} \\ = 6 \end{array}$

$\begin{array}{l} \sqrt{\frac{3}{2}} \times \sqrt{\frac{8}{27}} \end{array}$

$\begin{array}{l} = \sqrt{\frac{3}{2} \times \frac{8}{27}} \\ = \sqrt{\frac{4}{9}} \end{array}$

$\begin{array}{l} = \sqrt{{(\frac{2}{3})}^{2}} \\ = \frac{2}{3} \end{array}$

$\begin{array}{l} \sqrt[3]{0 / 01} \times \sqrt[3]{0 / 8} \end{array}$

$\begin{array}{l} = \sqrt[3]{0 / 01 \times 0 / 8} \\ = \sqrt[3]{0 / 008} \end{array}$

$\begin{array}{l} = \sqrt[3]{{(0 / 2)}^{3}} \\ = 0 / 2 \end{array}$

$\begin{array}{l} 3 \sqrt{2} \times 2 \sqrt{3} \end{array}$

$\begin{array}{l} = (3 \times 2) \times \sqrt{2 \times 3} \\ = 6 \sqrt{6} \end{array}$

$\begin{array}{l} - 3 \sqrt{5} \times (- 2 \sqrt{20}) \end{array}$

$\begin{array}{l} = [(- 3) \times (- 2)] \times \sqrt{5 \times 20} \\ = 6 \sqrt{100} \end{array}$

$\begin{array}{l} = 6 \sqrt{{(10)}^{2}} \\ = 6 \times 10 \\ = 60 \end{array}$

$\begin{array}{l} \sqrt{2} \times \sqrt{18} \end{array}$

$\begin{array}{l} = \sqrt{2 \times 18} \\ = \sqrt{36} \\ = \sqrt{6^{2}} \\ = 6 \end{array}$

$\begin{array}{l} \sqrt{\frac{2}{3}} \times \sqrt{6} \times \sqrt{\frac{5}{7}} \times \sqrt{14} \times \sqrt{10} \end{array}$

$\begin{array}{l} = \sqrt{\frac{2}{3} \times 6 \times \frac{5}{7} \times 14 \times 10} \\ = \sqrt{2 \times 2 \times 5 \times 2 \times 10} \end{array}$

$\begin{array}{l} = \sqrt{20 \times 20} \\ = \sqrt{20^{2}} \\ = 20 \end{array}$

$\begin{array}{l} 2 \sqrt[3]{a^{4} b^{3}} \times \sqrt[3]{a^{2} b} \end{array}$

$\begin{array}{l} = 2 \sqrt[3]{(a^{4} b^{3}) \times (a^{2} b)} \\ = 2 \sqrt[3]{(a^{6} b^{3}) \times b} \end{array}$

$\begin{array}{l} = 2 \sqrt[3]{{(a^{2} b)}^{3}} \times \sqrt[3]{b} \\ = 2 a^{2} b \sqrt[3]{b} \end{array}$

$\begin{array}{l} 2 \sqrt[3]{4} \times (- 3) \sqrt[3]{2} \end{array}$

$\begin{array}{l} = [2 \times (- 3)] \times \sqrt[3]{4 \times 2} \\ = - 6 \sqrt[3]{8} \end{array}$

$\begin{array}{l} = - 6 \sqrt[3]{{(2)}^{3}} \\ = - 6 \times 2 \\ = - 12 \end{array}$

$\begin{array}{l} \sqrt[3]{2 b^{2}} \times \sqrt[3]{4 b^{4}} \end{array}$

$\begin{array}{l} = \sqrt[3]{(2 b^{2}) \times (4 b^{4})} \\ = \sqrt[3]{8 b^{6}} \end{array}$

$\begin{array}{l} = \sqrt[3]{{(2 b^{2})}^{3}} \\ = 2 b^{2} \end{array}$

تمرین

عبارات راديكالی زير را محاسبه کنید.

$\begin{array}{l} \sqrt[3]{- 16} \end{array}$

$\begin{array}{l} = \sqrt[3]{- 8 \times 2} \\ = \sqrt[3]{- 8} \times \sqrt[3]{2} \end{array}$

$\begin{array}{l} = \sqrt[3]{{(- 2)}^{3}} \times \sqrt[3]{2} \\ = - 2 \sqrt[3]{2} \end{array}$

$\begin{array}{l} \sqrt[3]{54} \end{array}$

$\begin{array}{l} = \sqrt[3]{27 \times 2} \\ = \sqrt[3]{27} \times \sqrt[3]{2} \end{array}$

$\begin{array}{l} = \sqrt[3]{{(3)}^{3}} \times \sqrt[3]{2} \\ = 3 \sqrt[3]{2} \end{array}$

$\begin{array}{l} 2 \sqrt{150} \end{array}$

$\begin{array}{l} = 2 \times \sqrt{25 \times 6} \\ = 2 \times \sqrt{25} \times \sqrt{6} \end{array}$

$\begin{array}{l} = 2 \times 5 \times \sqrt{6} \\ = 10 \sqrt{6} \end{array}$

$\begin{array}{l} \sqrt[3]{0 / 003} \end{array}$

$\begin{array}{l} = \sqrt[3]{3 \times (0 / 001)} \\ = \sqrt[3]{3} \times \sqrt[3]{0 / 001} \end{array}$

$\begin{array}{l} = \sqrt[3]{3} \times \sqrt[3]{{(0 / 1)}^{3}} \\ = \sqrt[3]{3} \times (0 / 1) \\ = 0 / 1 \sqrt[3]{3} \end{array}$

$\begin{array}{l} \sqrt{128 a^{5}} \end{array}$

$\begin{array}{l} = \sqrt{(2 \times 64) \times (a^{4} \times a)} \\ = \sqrt{64 \times a^{4}} \times \sqrt{2 \times a} \end{array}$

$\begin{array}{l} = (8 \times a^{2}) \times \sqrt{2 a} \\ = 8 a^{2} \sqrt{2 a} \end{array}$

تمرین

درستی تساوی‌ های زیر را نشان دهید.

$\sqrt{2} \times \sqrt{32} = 8$

$\begin{array}{l} \sqrt{2} \times \sqrt{32} \\ = \sqrt{2 \times 32} \end{array}$

$\begin{array}{l} = \sqrt{64} \\ = 8 \end{array}$

$\sqrt[3]{- 9} \times \sqrt[3]{3} = - 3$

$\begin{array}{l} \sqrt[3]{- 9} \times \sqrt[3]{3} \\ = \sqrt[3]{- 9 \times 3} \end{array}$

$\begin{array}{l} = \sqrt[3]{- 27} \\ = - 3 \end{array}$

$\sqrt[3]{b a^{2}} \times \sqrt[3]{a b^{2}} = a b$

$\begin{array}{l} \sqrt[3]{b a^{2}} \times \sqrt[3]{a b^{2}} \\ = \sqrt[3]{(b a^{2}) (a b^{2})} \end{array}$

$\begin{array}{l} = \sqrt[3]{a^{3} . b^{3}} \\ = \sqrt[3]{{(a b)}^{3}} \\ = a b \end{array}$

${(\sqrt[3]{a})}^{2} = \sqrt[3]{a^{2}}$

$\begin{array}{l} {(\sqrt[3]{a})}^{2} \\ = (\sqrt[3]{a}) (\sqrt[3]{a}) \end{array}$

$\begin{array}{l} = \sqrt[3]{a . a} \\ = \sqrt[3]{a^{2}} \end{array}$

$\begin{array}{l} \sqrt{0 / 2} \times \sqrt{0 / 18} \end{array} = \frac{6}{10} \sqrt{\frac{1}{10}}$

$\begin{array}{l} \begin{array}{l} \sqrt{0 / 2} \times \sqrt{0 / 18} \end{array} \\ \begin{array}{l} = \sqrt{\frac{2}{10} \times \frac{18}{100}} \\ = \sqrt{\frac{36}{1000}} \end{array} \end{array}$

$\begin{array}{l} = \sqrt{\frac{1}{10} \times \frac{36}{100}} \\ = \sqrt{\frac{1}{10} \times {(\frac{6}{10})}^{2}} \\ = \frac{6}{10} \sqrt{\frac{1}{10}} \end{array}$

$\begin{array}{l} - 5 \sqrt{6} \times (- 2) \sqrt{18} \end{array} = 60 \sqrt{3}$

$\begin{array}{l} \begin{array}{l} - 5 \sqrt{6} \times (- 2) \sqrt{18} \end{array} \\ = [(- 5) \times (- 2)] \times \sqrt{6 \times 18} \\ = 10 \sqrt{108} \\ = 10 \sqrt{36 \times 3} \end{array}$

$\begin{array}{l} = 10 \times \sqrt{36} \times \sqrt{3} \\ = 10 \times \sqrt{{(6)}^{2}} \times \sqrt{3} \\ = 10 \times 6 \times \sqrt{3} \\ = 60 \sqrt{3} \end{array}$

$\begin{array}{l} - 3 \sqrt{5} \times (- 2 \sqrt{20}) \end{array} = 60$

$\begin{array}{l} \begin{array}{l} - 3 \sqrt{5} \times (- 2 \sqrt{20}) \end{array} \\ = [(- 3) \times (- 2)] \times \sqrt{5 \times 20} \\ = 6 \sqrt{100} \end{array}$

$\begin{array}{l} = 6 \sqrt{{(10)}^{2}} \\ = 6 \times 10 \\ = 60 \end{array}$

$\begin{array}{l} \sqrt[3]{- 2} \times \sqrt[3]{5} \end{array} = \sqrt[3]{- 10}$

$\begin{array}{l} \begin{array}{l} \sqrt[3]{- 2} \times \sqrt[3]{5} \end{array} \\ = \sqrt[3]{(- 2) \times 5} \\ = \sqrt[3]{- 10} \end{array}$

$\begin{array}{l} \sqrt{14} \times \sqrt{7} \times \sqrt{2} \end{array} = 14$

$\begin{array}{l} \begin{array}{l} \sqrt{14} \times \sqrt{7} \times \sqrt{2} \end{array} \\ = \sqrt{14 \times 7 \times 2} \\ = \sqrt{14 \times 14} \end{array}$

$\begin{array}{l} = \sqrt{14^{2}} \\ = 14 \end{array}$

$\begin{array}{l} \sqrt[3]{9 - \sqrt{17}} \times \sqrt[3]{9 + \sqrt{17}} \end{array} = 4$

$\begin{array}{l} \begin{array}{l} \sqrt[3]{9 - \sqrt{17}} \times \sqrt[3]{9 + \sqrt{17}} \end{array} \\ = \sqrt[3]{(9 - \sqrt{17}) \times (9 + \sqrt{17})} \\ = \sqrt[3]{{(9)}^{2} - {(\sqrt{17})}^{2}} \end{array}$

$\begin{array}{l} = \sqrt[3]{81 - 17} \\ = \sqrt[3]{64} \\ = \sqrt[3]{4^{3}} \\ = 4 \end{array}$

$\begin{array}{l} \sqrt[5]{3} \times \sqrt[5]{9} \times \sqrt[5]{\frac{3}{5}} \times \sqrt[5]{\frac{5}{9}} \times \sqrt[5]{27} \end{array} = 3$

$\begin{array}{l} \begin{array}{l} \sqrt[5]{3} \times \sqrt[5]{9} \times \sqrt[5]{\frac{3}{5}} \times \sqrt[5]{\frac{5}{9}} \times \sqrt[5]{27} \end{array} \\ = \sqrt[5]{3 \times 9 \times \frac{3}{5} \times \frac{5}{9} \times 27} \\ = \sqrt[5]{9 \times 27} \end{array}$

$\begin{array}{l} = \sqrt[5]{3^{2} \times 3^{3}} \\ = \sqrt[5]{3^{5}} \\ = 3 \end{array}$

$\begin{array}{l} 2 \sqrt{a^{4} b} \times 3 \sqrt{a^{2} b^{3}} \end{array} = 6 a^{3} b^{2}$

$\begin{array}{l} \begin{array}{l} 2 \sqrt{a^{4} b} \times 3 \sqrt{a^{2} b^{3}} \end{array} \\ = (2 \times 3) \times \sqrt{(a^{4} b) \times (a^{2} b^{3})} \end{array}$

$\begin{array}{l} = 6 \sqrt{a^{6} b^{4}} \\ = 6 \sqrt{{(a^{3} b^{2})}^{2}} \\ = 6 a^{3} b^{2} \end{array}$

$\begin{array}{l} \sqrt[3]{a b c} \times \sqrt[3]{a^{2} b^{5} c^{8}} \end{array} = a b^{2} c^{3}$

$\begin{array}{l} \begin{array}{l} \sqrt[3]{a b c} \times \sqrt[3]{a^{2} b^{5} c^{8}} \end{array} \\ = \sqrt[3]{a b c \times a^{2} b^{5} c^{8}} \end{array}$

$\begin{array}{l} = \sqrt[3]{a^{3} \times b^{6} \times c^{9}} \\ = \sqrt[3]{{(a b^{2} c^{3})}^{3}} \\ = a b^{2} c^{3} \end{array}$

$\begin{array}{l} 2 \sqrt[3]{a^{4} b^{3}} \times \sqrt[3]{a^{2} b} \end{array} = 2 a^{2} b \sqrt[3]{b}$

$\begin{array}{l} \begin{array}{l} 2 \sqrt[3]{a^{4} b^{3}} \times \sqrt[3]{a^{2} b} \end{array} \\ = 2 \sqrt[3]{(a^{4} b^{3}) \times (a^{2} b)} \end{array}$

$\begin{array}{l} = 2 \sqrt[3]{(a^{6} b^{3}) \times b} \\ = 2 \sqrt[3]{{(a^{2} b)}^{3}} \times \sqrt[3]{b} \\ = 2 a^{2} b \sqrt[3]{b} \end{array}$

$\begin{array}{l} \sqrt[3]{a^{2} + b^{2}} \times \sqrt[3]{a^{4} + 2 a^{2} b^{2} + b^{4}} \end{array} = a^{2} + b^{2}$

$\begin{array}{l} \begin{array}{l} \sqrt[3]{a^{2} + b^{2}} \times \sqrt[3]{a^{4} + 2 a^{2} b^{2} + b^{4}} \end{array} \\ = \sqrt[3]{(a^{2} + b^{2}) (a^{4} + 2 a^{2} b^{2} + b^{4})} \end{array}$

$\begin{array}{l} = \sqrt[3]{(a^{2} + b^{2}) {(a^{2} + b^{2})}^{2}} \\ = \sqrt[3]{{(a^{2} + b^{2})}^{3}} \\ = a^{2} + b^{2} \end{array}$

$\sqrt[3]{a^{2} b c} \times \sqrt[3]{a b^{2} c} \times \sqrt[3]{c} = a b c$

$\begin{array}{l} \sqrt[3]{a^{2} b c} \times \sqrt[3]{a b^{2} c} \times \sqrt[3]{c} \\ = \sqrt[3]{a^{2} b c \times a b^{2} c \times c} \end{array}$

$\begin{array}{l} = \sqrt[3]{a^{3} b^{3} c^{3}} \\ = \sqrt[3]{{(a b c)}^{3}} \\ = a b c \end{array}$

$\sqrt[5]{a^{2} b^{3}} \times \sqrt[5]{a^{3} b^{2}} = a b$

$\begin{array}{l} \sqrt[5]{a^{2} b^{3}} \times \sqrt[5]{a^{3} b^{2}} \\ = \sqrt[5]{a^{2} b^{3} \times a^{3} b^{2}} \end{array}$

$\begin{array}{l} = \sqrt[5]{a^{5} b^{5}} \\ = \sqrt[5]{{(a b)}^{5}} \\ = a b \end{array}$

$\sqrt[7]{a^{2} b^{3} c} \times \sqrt[7]{a^{3} b^{2} c^{4}} \times \sqrt[7]{a^{2} b^{2} c^{2}} = a b c$

$\begin{array}{l} \sqrt[7]{a^{2} b^{3} c} \times \sqrt[7]{a^{3} b^{2} c^{4}} \times \sqrt[7]{a^{2} b^{2} c^{2}} \\ = \sqrt[7]{a^{2} b^{3} c \times a^{3} b^{2} c^{4} \times a^{2} b^{2} c^{2}} \end{array}$

$\begin{array}{l} = \sqrt[7]{a^{7} b^{7} c^{7}} \\ = \sqrt[7]{{(a b c)}^{7}} \\ = a b c \end{array}$

$\sqrt[3]{a - b} \times \sqrt[3]{a^{2} + a b + b^{2}} \times \sqrt[3]{{(a^{3} - b^{3})}^{2}} = (a^{3} - b^{3})$

$\begin{array}{l} \sqrt[3]{a - b} \times \sqrt[3]{a^{2} + a b + b^{2}} \times \sqrt[3]{{(a^{3} - b^{3})}^{2}} \end{array}$

$\begin{array}{l} = \sqrt[3]{(a - b) (a^{2} + a b + b^{2}) {(a^{3} - b^{3})}^{2}} \end{array}$

$\begin{array}{l} = \sqrt[3]{(a^{3} - b^{3}) {(a^{3} - b^{3})}^{2}} \\ = \sqrt[3]{{(a^{3} - b^{3})}^{3}} \\ = (a^{3} - b^{3}) \end{array}$

$\sqrt[4]{\sqrt{2} - 1} \times \sqrt[4]{\sqrt{2} + 1} = 1$

$\begin{array}{l} \sqrt[4]{\sqrt{2} - 1} \times \sqrt[4]{\sqrt{2} + 1} \\ = \sqrt[4]{(\sqrt{2} - 1) (\sqrt{2} + 1)} \\ = \sqrt[4]{{(\sqrt{2})}^{2} - {(1)}^{2}} \end{array}$

$\begin{array}{l} = \sqrt[4]{2 - 1} \\ = \sqrt[4]{1} \\ = 1 \end{array}$

$\sqrt[4]{x^{4} - x^{2} + 1} \times \sqrt[4]{(x^{2} + 1) {(x^{6} + 1)}^{2}} = \sqrt[4]{{(x^{6} + 1)}^{3}}$

$\begin{array}{l} \sqrt[4]{x^{4} - x^{2} + 1} \times \sqrt[4]{(x^{2} + 1) {(x^{6} + 1)}^{2}} \end{array}$

$\begin{array}{l} = \sqrt[4]{(x^{2} + 1) (x^{4} - x^{2} + 1) {(x^{6} + 1)}^{2}} \end{array}$

$\begin{array}{l} = \sqrt[4]{(x^{6} + 1) {(x^{6} + 1)}^{2}} \\ = \sqrt[4]{{(x^{6} + 1)}^{3}} \end{array}$

$\sqrt[7]{a^{4} b^{5}} \times \sqrt[7]{a^{3} b c^{3}} \times \sqrt[7]{b c^{4}} = a b c$

$\begin{array}{l} \sqrt[7]{a^{4} b^{5}} \times \sqrt[7]{a^{3} b c^{3}} \times \sqrt[7]{b c^{4}} \\ = \sqrt[7]{a^{4} b^{5} \times a^{3} b c^{3} \times b c^{4}} \end{array}$

$\begin{array}{l} = \sqrt[7]{a^{7} \times b^{7} \times c^{7}} \\ = \sqrt[7]{{(a b c)}^{7}} \\ = a b c \end{array}$

$\frac{\sqrt[5]{30} \times \sqrt[5]{16}}{\sqrt[5]{3} \times \sqrt[5]{5}} = 2$

$\begin{array}{l} \frac{\sqrt[5]{30} \times \sqrt[5]{16}}{\sqrt[5]{3} \times \sqrt[5]{5}} \\ = \frac{\sqrt[5]{30 \times 16}}{\sqrt[5]{3 \times 5}} \end{array}$

$\begin{array}{l} = \sqrt[5]{\frac{30 \times 16}{3 \times 5}} \\ = \sqrt[5]{\frac{32}{1}} \\ = \sqrt[5]{2^{5}} \\ = 2 \end{array}$

${(\sqrt{3} - \sqrt{2})}^{2} (5 + 2 \sqrt{6}) = 1$

$\begin{array}{l} {(\sqrt{3} - \sqrt{2})}^{2} (5 + 2 \sqrt{6}) \\ = (3 - 2 \sqrt{3} \sqrt{2} + 2) (5 + 2 \sqrt{6}) \\ = (5 - 2 \sqrt{6}) (5 + 2 \sqrt{6}) \end{array}$

$\begin{array}{l} = {(5)}^{2} - {(2 \sqrt{6})}^{2} \\ = (25) - (4 \times 6) \\ = 1 \end{array}$

${(\sqrt{3} - \sqrt{5})}^{2} + 2 \sqrt{15} = 8$

$\begin{array}{l} {(\sqrt{3} - \sqrt{5})}^{2} + 2 \sqrt{15} \\ = (3 - 2 \sqrt{3} \sqrt{5} + 5) + 2 \sqrt{15} \end{array}$

$\begin{array}{l} = (8 - 2 \sqrt{15}) + 2 \sqrt{15} \\ = 8 \end{array}$

${(\sqrt{3} - \sqrt{7})}^{2} (10 - 2 \sqrt{21}) {(10 + 2 \sqrt{21})}^{2} = 256$

$\begin{array}{l} {(\sqrt{3} - \sqrt{7})}^{2} (10 - 2 \sqrt{21}) {(10 + 2 \sqrt{21})}^{2} \end{array}$

$\begin{array}{l} = (3 - 2 \sqrt{3} \sqrt{7} + 7) (10 - 2 \sqrt{21}) {(10 + 2 \sqrt{21})}^{2} \end{array}$

$= (10 - 2 \sqrt{21}) (10 - 2 \sqrt{21}) {(10 + 2 \sqrt{21})}^{2}$

$\begin{array}{l} = {(10 - 2 \sqrt{21})}^{2} {(10 + 2 \sqrt{21})}^{2} \\ = {[(10 - 2 \sqrt{21}) (10 + 2 \sqrt{21})]}^{2} \\ = {[100 - {(2 \sqrt{21})}^{2}]}^{2} \end{array}$

$\begin{array}{l} = {(100 - 4 \times 21)}^{2} \\ = {(100 - 84)}^{2} \\ = {(16)}^{2} \\ = 256 \end{array}$

$\sqrt{2 + \sqrt{3}} \times \sqrt{2 + \sqrt{2 + \sqrt{3}}} \times \sqrt{2 + \sqrt{2 + \sqrt{2 + \sqrt{3}}}} \times \sqrt{2 - \sqrt{2 + \sqrt{2 + \sqrt{3}}}} = 1$

$\sqrt{2 + \sqrt{3}} \times \sqrt{2 + \sqrt{2 + \sqrt{3}}} \times \sqrt{2 + \sqrt{2 + \sqrt{2 + \sqrt{3}}}} \times \sqrt{2 - \sqrt{2 + \sqrt{2 + \sqrt{3}}}}$

$\begin{array}{l} = \sqrt{2 + \sqrt{3}} \times \sqrt{2 + \sqrt{2 + \sqrt{3}}} \times [(\sqrt{2 + \sqrt{2 + \sqrt{2 + \sqrt{3}}}}) \times (\sqrt{2 - \sqrt{2 + \sqrt{2 + \sqrt{3}}}})] \end{array}$

$\begin{array}{l} = \sqrt{2 + \sqrt{3}} \times \sqrt{2 + \sqrt{2 + \sqrt{3}}} \times [\sqrt{(2 + \sqrt{2 + \sqrt{2 + \sqrt{3}}}) \times (2 - \sqrt{2 + \sqrt{2 + \sqrt{3}}})}] \end{array}$

$\begin{array}{l} = \sqrt{2 + \sqrt{3}} \times \sqrt{2 + \sqrt{2 + \sqrt{3}}} \times (\sqrt{4 - (2 + \sqrt{2 + \sqrt{3}}}) \end{array}$

$\begin{array}{l} = \sqrt{2 + \sqrt{3}} \times \sqrt{2 + \sqrt{2 + \sqrt{3}}} \times \sqrt{2 - \sqrt{2 + \sqrt{3}}} \end{array}$

$\begin{array}{l} = \sqrt{2 + \sqrt{3}} \times \sqrt{4 - (2 + \sqrt{3})} = \sqrt{2 + \sqrt{3}} \times \sqrt{2 - \sqrt{3}} \end{array}$

$\begin{array}{l} = \sqrt{4 - 3} \\ = 1 \end{array}$

تمرین

مجموع ارقام حاصل عبارت زیر را بیابید.

$A = \sqrt{1.111.111.111 - 22.222}$

$\begin{array}{l} \Rightarrow A = \sqrt{(1.111.100.000 + 11.111) - 2 (11.111)}; a = 11.111 \end{array}$

$\begin{array}{l} \Rightarrow A = \sqrt{(a 00.000 + a) - 2 (a)} \\ \Rightarrow A = \sqrt{a 00.001 - 2 a} \\ \Rightarrow A = \sqrt{99999 a} \end{array}$

$\begin{array}{l} \Rightarrow A = \sqrt{9 (11.111) a}; a = 11.111 \\ \Rightarrow A = \sqrt{9 (a) a} \\ \Rightarrow A = \sqrt{9 a^{2}} \end{array}$

$\begin{array}{l} \Rightarrow A = 3 a \\ \Rightarrow A = 3 (11.111) \\ \Rightarrow A = 33.333 \end{array}$

مجموع ارقام $15$ می‌باشد.

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نکته

در قضیه فوق بیان کردیم $\sqrt[n]{a} . \sqrt[n]{b} = \sqrt[n]{a b}$

عكس قضیه فوق در حالت كلی برقرار نیست، یعنی تساوی $\sqrt[n]{a b} = \sqrt[n]{a} . \sqrt[n]{b}$ همواره برقرار نیست.

به‌عنوان نمونه:

$\sqrt{(- 2) (- 3)} \neq \sqrt{- 2} \times \sqrt{- 3}$

تمرین

درستی تساوی های زیر را بررسی کنید.

$\sqrt{300} = 10 \sqrt{3}$

$\begin{array}{l} \sqrt{300} \\ = \sqrt{100 \times 3} \\ = \sqrt{100} \times \sqrt{3} \\ = 10 \sqrt{3} \end{array}$

$\begin{array}{l} \sqrt[4]{243} = 3 \sqrt[4]{3} \end{array}$

$\begin{array}{l} \sqrt[4]{243} \\ = \sqrt[4]{81 \times 3} \\ = \sqrt[4]{81} \times \sqrt[4]{3} \\ = \sqrt[4]{3^{4}} \times \sqrt[4]{3} \\ = 3 \sqrt[4]{3} \end{array}$

$\begin{array}{l} \sqrt[3]{{(a + b)}^{4}} = (a + b) \sqrt[3]{a + b} \end{array}$

$\begin{array}{l} \sqrt[3]{{(a + b)}^{4}} \\ = \sqrt[3]{{(a + b)}^{3} \times (a + b)} \\ = \sqrt[3]{{(a + b)}^{3}} \times \sqrt[3]{(a + b)} \\ = (a + b) \sqrt[3]{a + b} \end{array}$

$\begin{array}{l} \sqrt[5]{a^{7} \times b^{6} \times c^{10}} = a b c^{2} \sqrt[5]{a^{2} \times b} \end{array}$

$\begin{array}{l} \sqrt[5]{a^{7} \times b^{6} \times c^{10}} \\ = \sqrt[5]{a^{5} \times a^{2} \times b^{5} \times b \times {(c^{2})}^{5}} \\ = \sqrt[5]{a^{5} \times b^{5} \times {(c^{2})}^{5}} \times \sqrt[5]{a^{2} \times b} \\ = a b c^{2} \sqrt[5]{a^{2} \times b} \end{array}$

$\sqrt[70]{x^{74} \times y^{75}} = x y \sqrt[70]{x^{4} \times y^{5}}$

$\begin{array}{l} \sqrt[70]{x^{74} \times y^{75}} \\ = \sqrt[70]{x^{70} \times x^{4} \times y^{70} \times y^{5}} \end{array}$

$\begin{array}{l} = \sqrt[70]{x^{70} \times y^{70}} \times \sqrt[70]{x^{4} \times y^{5}} \\ = x y \sqrt[70]{x^{4} \times y^{5}} \end{array}$

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