Ответ:

Объяснение:

1)  (a^(1/2))^1/3 = a^(1/2-1/3) = a^(1/6) - (корень 6 степени из a);

2)  (a^1/3)^1/4 =a^(1/3*1/4) = a^(1/12);

3) (16^1/5)^1/4 = (2⁴)^(1/5*1/4) = 2^(4/20) =2^(1/5);

4)  (27^1/4)^1/3 = (3³)^(1/4*1/3) = 3^(3/12) = 3^(1/4);

5)  (a*a^(1/2)^(1/3) = (a^(1+1/2))^1/3 = (a^(3/2)^1/3 = a^(3/2 * 1/3) = a^(1/2);

6)  (b*b^(1/2))^(1/6) = (b^(1+1/2))^(1/6) = (b^(3/2)^1/6 = b^(3/2*1/6) = b^(1/4);

7)  (1/a³*a^(1/3))^(1/4) = (a^(-3)*a^(1/3))^(1/4) = a^(-3+1/3)^(1/4) = a^(-8/3)^(1/4)=

=a^(-8/3*1/4) = a^(-2/3) = a^(3/2);

8)  ((1/x^3)*(1/x)^(1/3))^(1/5) = (x^(-3)*(x)^(-1/3))^(1/5)= (x^(-3+(-1/3))^(1/5) =

= x^(-10/3 * 1/5) = x^(-2/3) = x^(3/2).

Verified answer

Формулы для решения:

[tex]\sqrt[n]{\sqrt[m]{x} }= \sqrt[n\times m]{x} \\\\\sqrt[n]{x^m} =x^{\frac{m}{n} }\\\\\sqrt{\dfrac{x}{y} } =\dfrac{\sqrt{x} }{\sqrt{y} } \\\\x^m\times x^n=x^{m+n}[/tex]

1)

[tex]\sqrt[3]{\sqrt{a} } =\sqrt[3\times2]{a} =\sqrt[6]{a}[/tex]

2)

[tex]\sqrt[4]{\sqrt[3]{a} } =\sqrt[4\times3]{a} =\sqrt[12]{a}[/tex]

3)

[tex]\sqrt[4]{\sqrt[5]{16} } =\sqrt[4\times5]{16} =\sqrt[20]{16}=\sqrt[20]{2^4}=\sqrt[5]{2}[/tex]

4)

[tex]\sqrt[3]{\sqrt[4]{27} } =\sqrt[3\times4]{27} =\sqrt[12]{27}=\sqrt[12]{3^3} =\sqrt[4]{3}[/tex]

5)

[tex]\sqrt[3]{a\sqrt{a} } =\sqrt[3]{\sqrt{a^2} \sqrt{a} } =\sqrt[3]{\sqrt{a^3} } =\sqrt[3\times2]{a^3}=\sqrt[6]{a^3} =\sqrt{a}[/tex]

6)

[tex]\sqrt[6]{b\sqrt{b} } =\sqrt[6]{\sqrt{b^2} \sqrt{b} } =\sqrt[6]{\sqrt{b^3} } =\sqrt[6\times2]{b^3}=\sqrt[12]{b^3} =\sqrt[4]{b}[/tex]

7)

[tex]\displaystyle\sqrt[4]{\frac{1}{a^3} \times\sqrt[3]{a} } =\sqrt[4]{\frac{\sqrt[3]{a}}{a^3}} =\frac{\sqrt[4]{\sqrt[3]{a} } }{\sqrt[4]{a^3} } =\frac{\sqrt[12]{a} }{{\sqrt[4]{a^3}}} =\frac{a^{\frac{1}{12} }}{a^{\frac{3}{4} }}=\frac{1}{a^{\frac{3}{4} -\frac{1}{12} }} =\frac{1}{a^{\frac{2}{3} }} =\frac{1}{\sqrt[3]{a^2} }[/tex]

8)

[tex]\displaystyle\sqrt[5]{\frac{1}{x^3} \times\sqrt[3]{\frac{1}{x} } } =\sqrt[5]{\frac{1}{x^3} \times x^{-\frac{1}{3} }} =\sqrt[5]{\frac{1}{x^{3-(-\frac{1}{3}) }} } =\sqrt[5]{\frac{1}{x^{\frac{10}{3} }} } =\sqrt[5]{\frac{1}{\sqrt[3]{x^{10}} } } =\frac{1}{\sqrt[5]{\sqrt[3]{x^{10}} } } =\\\\=\frac{1}{\sqrt[15]{x^{10}} } =\frac{1}{\sqrt[3]{x^2} }[/tex]