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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 antiderivative of $\sqrt{u}$ is found by rewriting the square root as a fractional power, $u^{1/2}$, and then applying the power rule for integration. The power rule states that $\int u^n du = \frac{u^{n+1}}{n+1}$.

Conditions: The integrand is $\sqrt{u}$, which is equivalent to $u^{1/2}$.; The power rule for integration is applicable.; $u \ge 0$ for the real-valued square root form.

The antiderivative of $\sqrt{u}$ is found by rewriting the square root as a fractional power, $u^{1/2}$, and then applying the power rule for integration. The power rule states that $\int u^n du = \frac{u^{n+1}}{n+1}$.

Conditions: The integrand is $\sqrt{u}$, which is equivalent to $u^{1/2}$.; The power rule for integration is applicable.; $u \ge 0$ for the real-valued square root form.

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 $\frac{2}{3}$.