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I want to understand how to convert a radical to a fractional exponent. Given the following equation:

$\sqrt[3]{(x)^6\cdot x^9}=\sqrt[3]{x^{24}\cdot x^9}=\sqrt[3]{x^{33}}=x^{\frac{33}3}=x^{11}$

How does: $\sqrt[3]{(x)^6\cdot x^9} = \sqrt[3]{x^{24}\cdot x^9}\;\;$??

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    $\begingroup$ $(x^4)^6 = \underbrace{x^4 x^4 \cdots x^4}_6 = \underbrace{(xxxx)(xxxx) \cdots (xxxx)}_6 = \underbrace{xxxxxxxx \cdots xxxx}_{24} = x^{24}$. $\endgroup$ – azif00 Apr 13 at 20:33
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    $\begingroup$ My original equation was changed from $\sqrt[3]{(x)^6 * x^9}$ to $\sqrt[3]{(x^4)^6 * x^9}$ . Why was the original equation modified. Are these expressions equal? $\endgroup$ – Untapped Insights Apr 13 at 21:44
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    $\begingroup$ There's a typo in your textbook. See the question "How did $(x)^6x^9$ become $x^{24}x^9$ in this textbook example?". $\endgroup$ – Blue Apr 13 at 22:12
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    $\begingroup$ "Why was the original equation modified. Are these expressions equal? " They are not. That is the problem. $\sqrt[3]{(x)^6 \cdot x^9} \ne \sqrt[3]{x^{24}\cdot x^9}$. The textbook you have had a typo and the question you asked is simply wrong. ... I assume the textbooks typo was the problem $\sqrt[3]{(x^{\color{red}4})^6\cdot 9}$ which will equal $\sqrt[3]{x^{24}\cdot x^9}$ for the standard rules of exponents. The person who edited fixed the typo figuring that that wasn't what was causing you trouble. $\endgroup$ – fleablood Apr 14 at 0:16
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Power rules of exponent :

  1. $(a^m)^n=a^{mn}$ $\to \sqrt[3]{(x^4)^6\cdot x^9} = \sqrt[3]{x^{24}\cdot x^9}$
  2. $a^ma^n=a^{m+n}$ $\to \sqrt[3]{x^{24}\cdot x^9}$$=\sqrt[3]{x^{33}}$
  3. $\sqrt[n]{a}=a^{\frac{1}{n}},\sqrt[n]{a^m}=a^{\frac{m}{n}} \to \sqrt[3]{x^{33}}$$=x^{\frac{33}{3}}=x^{11}\;$

Or,

$\sqrt[3]{(x^4)^6\cdot x^9} = \sqrt[3]{x^{24}\cdot x^9}$(2)$=\sqrt[3]{x^{33}}$$=x^{\frac{33}{3}}=x^{11}\;$

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    $\begingroup$ My original equation was changed from $\sqrt[3]{(x)^6 * x^9}$ to $\sqrt[3]{(x^4)^6 * x^9}$ . Why was the original equation modified. Are these expressions equal? Your answer makes sense to me in the event that the expression is: $\sqrt[3]{(x^4)^6 * x^9}$ . However it does not make sense to me that: $\sqrt[3]{(x)^6 * x^9}$ is equal... $\endgroup$ – Untapped Insights Apr 13 at 21:46
  • $\begingroup$ Don’t know who changed your original question. The two expressions are equal only when x=0, -1,+1 (given x is real). $\endgroup$ – BStar Apr 13 at 21:52
  • $\begingroup$ Thats what i thought. I modified my original question back to reflect what it was initially: $\sqrt[3]{(x)^6 * x^9}$ . $\endgroup$ – Untapped Insights Apr 13 at 21:56
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    $\begingroup$ It's possible that the textbook may contain a typo if in fact: $\sqrt[3]{(x)^6\cdot x^9}$ doesn't equal $\sqrt[3]{x^{24}\cdot x^9}\;\;$ $\endgroup$ – Untapped Insights Apr 13 at 21:58
  • $\begingroup$ Most likely, what matters is you caught its mistake:-). $\endgroup$ – BStar Apr 13 at 22:02
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The expressions you wrote are not equal for more or less the reasons you have identified, that $x^6 \neq x^{24}$.

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