Respuesta :

[tex]\bf a^{\frac{{ n}}{{ m}}} \implies \sqrt[{ m}]{a^{ n}} \qquad \qquad \sqrt[{ m}]{a^{ n}}\implies a^{\frac{{ n}}{{ m}}}\\\\ -------------------------------[/tex]

[tex]\bf \left( \cfrac{750}{512} \right)^{\frac{1}{3}}\implies \cfrac{750^{\frac{1}{3}}}{512^{\frac{1}{3}}}\implies \cfrac{\sqrt[3]{750^1}}{\sqrt[3]{512^1}}\quad \begin{cases} 750=2\cdot 3\cdot 5\cdot 5\cdot 5\\ \qquad 2\cdot 3\cdot 5^3\\ \qquad 6\cdot 5^3\\ 512=8\cdot 8\cdot 8\\ \qquad 8^3 \end{cases} \\\\\\ \cfrac{\sqrt[3]{750}}{\sqrt[3]{512}}\implies \cfrac{\sqrt[3]{6\cdot 5^3}}{\sqrt[3]{8^3}}\implies \cfrac{5\sqrt[3]{6}}{8}[/tex]

Answer:

[tex]\frac{5\sqrt[3]{6}}{8}[/tex]

Step-by-step explanation:

We have been given an exponential number. We are supposed to simplify our given number.

[tex](\frac{750}{512})^{\frac{1}{3}[/tex]  

Using fractional exponent rule [tex]x^\frac{m}{n}=\sqrt[n]{x^m}[/tex], w ecan write our given number as:

[tex](\frac{750}{512})^{\frac{1}{3}}=\sqrt[3]{(\frac{750}{512})^1}[/tex]

[tex](\frac{750}{512})^{\frac{1}{3}}=\sqrt[3]{\frac{750}{512}}[/tex]

We can rewrite 512 as [tex]8^3[/tex] and 750 as 125*6.

[tex](\frac{750}{512})^{\frac{1}{3}}=\sqrt[3]{\frac{125*6}{8^3}}[/tex]

We can rewrite 125 as [tex]5^3[/tex]

[tex](\frac{750}{512})^{\frac{1}{3}}=\sqrt[3]{\frac{5^3*6}{8^3}}[/tex]

Using radical rule [tex]\sqrt[n]{a^n}=a[/tex], we will get:

[tex](\frac{750}{512})^{\frac{1}{3}}=\frac{5\sqrt[3]{6}}{8}[/tex]

Therefore, [tex]\frac{5\sqrt[3]{6}}{8}[/tex] is equivalent to our given number.

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