Given x > 0, the expression sqrt(x^(8)) is equivalent to x^(4)sqrtx x^(5) x^(5)sqrtx x^(4)

Apply square root property: We are given the expression sqrt(x^(8)) and we need to simplify it. Since x > 0, we can apply the property of square roots that sqrt(a^2) = a for any non-negative a. In this case, we have a power of 8, which is an even number, so we can rewrite the expression as (x^(8))^(1/2).Use power rule of exponents: Using the power rule of exponents (a^(m*n) = (a^m)^n), we can simplify the expression (x^(8))^(1/2) to x^(8/2).Divide exponent by 2: Dividing the exponent 8 by 2, we get x^(4). This is because 8 divided by 2 equals 4.Final simplified form: Since x > 0, we do not need to consider the absolute value of x, and x^(4) is the simplified form of sqrt(x^(8)).

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Find indefinite integrals using the substitution

Full solution

Q. Given

x>0

, the expression

\sqrt{x^{8}}

is equivalent to

\newline

x^{4} \sqrt{x}

\newline

x^{5}

\newline

x^{5} \sqrt{x}

\newline

x^{4}

Apply square root property: We are given the expression $\sqrt{x^{8}}$ and we need to simplify it. Since x > 0, we can apply the property of square roots that $\sqrt{a^2} = a$ for any non-negative $a$ . In this case, we have a power of $8$ , which is an even number, so we can rewrite the expression as $(x^{8})^{\frac{1}{2}}$ .
Use power rule of exponents: Using the power rule of exponents $(a^{m*n} = (a^m)^n)$ , we can simplify the expression $(x^{8})^{1/2}$ to $x^{8/2}$ .
Divide exponent by $2$ : Dividing the exponent $8$ by $2$ , we get $x^{4}$ . This is because $8$ divided by $2$ equals $4$ .
Final simplified form: Since x > 0, we do not need to consider the absolute value of $x$ , and $x^{4}$ is the simplified form of $\sqrt{x^{8}}$ .