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The final exact value obtained for the displayed arc-length integral is $\frac{208}{27}$. This is calculated by multiplying the outside constant $\frac{8}{27}$ by the evaluated bracket difference 26.

Conditions: The integral has been fully substituted and evaluated up to the step $\frac{8}{27}\cdot 26$.; The arithmetic simplification $8 \cdot 26 = 208$ is performed exactly.

The final exact value obtained for the displayed arc-length integral is $\frac{208}{27}$. This is calculated by multiplying the outside constant $\frac{8}{27}$ by the evaluated bracket difference 26.

Conditions: The integral has been fully substituted and evaluated up to the step $\frac{8}{27}\cdot 26$.; The arithmetic simplification $8 \cdot 26 = 208$ is performed exactly.

The expression $9^{3/2}-1^{3/2}$ becomes 26 because $9^{3/2}$ evaluates to 27 and $1^{3/2}$ evaluates to 1. The power $9^{3/2}$ can be calculated by taking the square root of 9, which is 3, and then cubing it, resulting in $3^3 = 27$.

Conditions: The expression is evaluated exactly using integer powers.; $9^{3/2} = 27$ and $1^{3/2} = 1$.

The expression $9^{3/2}-1^{3/2}$ becomes 26 because $9^{3/2}$ evaluates to 27 and $1^{3/2}$ evaluates to 1. The power $9^{3/2}$ can be calculated by taking the square root of 9, which is 3, and then cubing it, resulting in $3^3 = 27$.

Conditions: The expression is evaluated exactly using integer powers.; $9^{3/2} = 27$ and $1^{3/2} = 1$.

The presenter factors out the common coefficient $\frac{2}{3}$ to make the subsequent arithmetic simplification easier. After evaluating the antiderivative at the upper and lower bounds, both terms inside the brackets contain the factor $\frac{2}{3}$.

Conditions: The expression to evaluate is $\frac{4}{9}\left[\frac{2}{3}\cdot 9^{3/2}-\frac{2}{3}\cdot 1^{3/2}\right]$.; Both bracketed terms contain the same factor $\frac{2}{3}$.

The presenter factors out the common coefficient $\frac{2}{3}$ to make the subsequent arithmetic simplification easier. After evaluating the antiderivative at the upper and lower bounds, both terms inside the brackets contain the factor $\frac{2}{3}$.

Conditions: The expression to evaluate is $\frac{4}{9}\left[\frac{2}{3}\cdot 9^{3/2}-\frac{2}{3}\cdot 1^{3/2}\right]$.; Both bracketed terms contain the same factor $\frac{2}{3}$.