which expression is equivalent to $sqrt4{x^{10}}$?\n$x^{2}(sqrt4{x^{2}})$\n$x^{2.2}$\n$x^{3}(sqrt4{x})$\n$x^{…

Explanation:

Step1: Rewrite the radical as an exponent

Use the rule $\sqrt[n]{a^m}=a^{\frac{m}{n}}$. So, $\sqrt[4]{x^{10}}=x^{\frac{10}{4}}$.

Step2: Simplify the exponent

$\frac{10}{4}=\frac{8 + 2}{4}=\frac{8}{4}+\frac{2}{4}=2+\frac{1}{2}$. Then $x^{\frac{10}{4}}=x^{2+\frac{1}{2}}=x^{2}\times x^{\frac{1}{2}}$. Also, $x^{\frac{1}{2}}=\sqrt{x}$, and we can rewrite $x^{\frac{10}{4}}$ as $x^{2}\sqrt[4]{x^{2}}$ since $x^{\frac{10}{4}}=x^{2+\frac{2}{4}}$ and using the property $a^{m + n}=a^{m}\times a^{n}$, where $a = x$, $m = 2$ and $n=\frac{2}{4}$.

Step3: Check each option

• Option 1: $x^{2}(\sqrt[4]{x^{2}})=x^{2}\times x^{\frac{2}{4}}=x^{2+\frac{2}{4}}=x^{\frac{8 + 2}{4}}=x^{\frac{10}{4}}=\sqrt[4]{x^{10}}$.
• Option 2: $x^{2.2}=x^{\frac{22}{10}}\neq x^{\frac{10}{4}}$.
• Option 3: $x^{3}(\sqrt[4]{x})=x^{3}\times x^{\frac{1}{4}}=x^{3+\frac{1}{4}}=x^{\frac{12 + 1}{4}}=x^{\frac{13}{4}}\neq x^{\frac{10}{4}}$.
• Option 4: $x^{5}\neq x^{\frac{10}{4}}$.

Answer:

$x^{2}(\sqrt[4]{x^{2}})$