Question

rewrite using a radical/root symbol:

1. ( x^{1/2} )
2. ( y^{\frac{5}{7}} )
3. ( 6^{\frac{10}{15}} )
4. ( 17490^{\frac{23}{78}} )

Explanation:

Step1: Apply rational exponent rule

Recall the rule: $a^{\frac{m}{n}} = \sqrt[n]{a^m}$

Step2: Rewrite each term with radicals

1. For $x^{\frac{1}{2}}$: $m=1$, $n=2$, so $\sqrt[2]{x^1} = \sqrt{x}$
2. For $y^{\frac{5}{7}}$: $m=5$, $n=7$, so $\sqrt[7]{y^5}$
3. For $6^{\frac{10}{15}}$: Simplify exponent first: $\frac{10}{15}=\frac{2}{3}$, then $\sqrt[3]{6^2}$
4. For $17490^{\frac{23}{78}}$: $m=23$, $n=78$, so $\sqrt[78]{17490^{23}}$

Answer:

1. $\sqrt{x}$
2. $\sqrt[7]{y^5}$
3. $\sqrt[3]{6^2}$ (or simplified form $\sqrt[3]{36}$)
4. $\sqrt[78]{17490^{23}}$