Given x > 0, the expression sqrt(x^(3)) is equivalent to x^(2) x x^(2)sqrt(x^(2)) xsqrtx

Rewrite using square root property: We need to simplify the expression sqrt(x^(3)). The square root of a number is the same as raising that number to the power of 1/2. Therefore, sqrt(x^(3)) is the same as (x^(3))^(1/2).Apply exponent property: Using the property of exponents that (a^(m))^(n) = a^(m*n), we can simplify (x^(3))^(1/2) to x^(3/2).Rewrite exponent: The expression x^(3/2) can be rewritten as x^(1+1/2), which is the same as x^(1)*x^(1/2).Final simplification: Since x^(1) is just x, and x^(1/2) is the square root of x, the expression x^(1)*x^(1/2) simplifies to x*sqrt(x).

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Find indefinite integrals using the substitution

Full solution

Q. Given

x>0

, the expression

\sqrt{x^{3}}

is equivalent to

\newline

x^{2}

\newline

x

\newline

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

\newline

x \sqrt{x}

Rewrite using square root property: We need to simplify the expression $\sqrt{x^{3}}$ . The square root of a number is the same as raising that number to the power of $\frac{1}{2}$ . Therefore, $\sqrt{x^{3}}$ is the same as $(x^{3})^{\frac{1}{2}}$ .
Apply exponent property: Using the property of exponents that $(a^{m})^{n} = a^{m*n}$ , we can simplify $(x^{3})^{\frac{1}{2}}$ to $x^{\frac{3}{2}}$ .
Rewrite exponent: The expression $x^{\frac{3}{2}}$ can be rewritten as $x^{1+\frac{1}{2}}$ , which is the same as $x^{1}\cdot x^{\frac{1}{2}}$ .
Final simplification: Since $x^{(1)}$ is just $x$ , and $x^{(1/2)}$ is the square root of $x$ , the expression $x^{(1)}*x^{(1/2)}$ simplifies to $x*\sqrt{x}$ .