# Converting a radical to a fractional exponent

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

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

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

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

Power rules of exponent :

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

Or,

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

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

The expressions you wrote are not equal for more or less the reasons you have identified, that $$x^6 \neq x^{24}$$.