Given x > 0, the expression sqrt(x^(21)) is equivalent to x^(11)sqrtx x^(11)sqrt(x^(2)) x^(10)sqrtx x^(10)sqrt(x^(2))

Understand sqrt(x): To simplify sqrt(x^(21)), we need to understand that sqrt(x) is the same as x^(1/2) and that we can use the property of exponents which states that (x^(a))^b = x^(a*b).Apply property of exponents: Applying the property of exponents to sqrt(x^(21)), we get (x^(21))^(1/2) = x^(21/2).Simplify exponent 21/2: The exponent 21/2 can be simplified by dividing 21 by 2, which gives us 10.5 or 10 + 1/2. Therefore, x^(21/2) is the same as x^(10) * x^(1/2).Rewrite as x^(10)sqrt(x): Since x^(1/2) is the same as sqrt(x), we can rewrite x^(10) * x^(1/2) as x^(10)sqrt(x).Check given options: Now we check the given options to see which one matches our simplified expression. The correct equivalent expression is x^(10)sqrt(x).

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Calculus

Find indefinite integrals using the substitution

Full solution

Q. Given

x>0

, the expression

\sqrt{x^{21}}

is equivalent to

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x^{11} \sqrt{x}

\newline

x^{11} \sqrt{x^{2}}

\newline

x^{10} \sqrt{x}

\newline

x^{10} \sqrt{x^{2}}

Understand $\sqrt{x}$ : To simplify $\sqrt{x^{21}}$ , we need to understand that $\sqrt{x}$ is the same as $x^{1/2}$ and that we can use the property of exponents which states that $(x^{a})^{b} = x^{a*b}$ .
Apply property of exponents: Applying the property of exponents to $\sqrt{x^{21}}$ , we get $(x^{21})^{1/2} = x^{\frac{21}{2}}$ .
Simplify exponent $\frac{21}{2}$ : The exponent $\frac{21}{2}$ can be simplified by dividing $21$ by $2$ , which gives us $10.5$ or $10 + \frac{1}{2}$ . Therefore, $x^{\frac{21}{2}}$ is the same as $x^{10} \cdot x^{\frac{1}{2}}$ .
Rewrite as $x^{10}\sqrt{x}$ : Since $x^{1/2}$ is the same as $\sqrt{x}$ , we can rewrite $x^{10} \times x^{1/2}$ as $x^{10}\sqrt{x}$ .
Check given options: Now we check the given options to see which one matches our simplified expression. The correct equivalent expression is $x^{10}\sqrt{x}$ .