Question

which expression is equivalent to \\(\sqrt{x^{10}}\\)?
options:
\\(x^5\\)
\\(x^{2.2}\\)
\\(x^2\left(\sqrt4{x^2}\
ight)\\)
\\(x^3\left(\sqrt4{x}\
ight)\\)

Explanation:

Step1: Simplify the original expression

We know that $\sqrt{x^{10}}$ can be written using the property of exponents $\sqrt{a^b}=a^{\frac{b}{2}}$. So for $\sqrt{x^{10}}$, we have $x^{\frac{10}{2}} = x^{5}$.

Step2: Analyze each option

• Option 1: $x^{5}$ is equal to our simplified expression $x^{5}$. Let's check other options to be sure.
• Option 2: $x^{3}(\sqrt[4]{x}) = x^{3}\times x^{\frac{1}{4}}=x^{3 + \frac{1}{4}}=x^{\frac{13}{4}} = x^{3.25}

eq x^{5}$.

• Option 3: $x^{2.2}=x^{\frac{11}{5}} = 2.2$, which is not equal to $x^{5}$.
• Option 4: $x^{2}(\sqrt[4]{x^{2}})=x^{2}\times x^{\frac{2}{4}}=x^{2+\frac{1}{2}}=x^{\frac{5}{2}} = x^{2.5}

eq x^{5}$.

Answer:

The expression equivalent to $\sqrt{x^{10}}$ is $x^{5}$ (the first option among the given choices, assuming the options are ordered as $x^{5}$, $x^{3}(\sqrt[4]{x})$, $x^{2.2}$, $x^{2}(\sqrt[4]{x^{2}})$). So the answer is the option with $x^{5}$.