$\left(2 \frac{2}{3} \times \frac{5}{8} \div 1 \frac{1}{4}\right)^{3}=$

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\left(2 \frac{2}{3} \times \frac{5}{8} \div 1 \frac{1}{4}\right)^{3}=

Simplify Inside Parentheses: Simplify the expression inside the parentheses. $\newline$ We have the expression $(2(\frac{2}{3})\times(\frac{5}{8})\div1(\frac{1}{4}))$ . We need to perform the multiplication and division in the correct order. $\newline$ First, we convert the mixed number $1(\frac{1}{4})$ to an improper fraction, which is $(4\times1 + 1)/4 = \frac{5}{4}$ .
Perform Multiplication: Perform the multiplication. $\newline$ Now we multiply $2\left(\frac{2}{3}\right)$ and $\left(\frac{5}{8}\right)$ . To do this, we first convert $2\left(\frac{2}{3}\right)$ to an improper fraction, which is $\left(3\cdot2 + 2\right)/3 = \frac{8}{3}$ . $\newline$ So, $\left(\frac{8}{3}\right) \times \left(\frac{5}{8}\right) = \left(8 \cdot 5\right) / \left(3 \cdot 8\right) = \frac{40}{24}$ . $\newline$ We can simplify this by dividing both numerator and denominator by $8$ , which gives us $\frac{5}{3}$ .
Perform Division: Perform the division. $\newline$ Now we divide the result from Step $2$ by $\frac{5}{4}$ . $\newline$ So, $(\frac{5}{3}) \div (\frac{5}{4}) = (\frac{5}{3}) \times (\frac{4}{5}) = (\frac{5 \times 4}{3 \times 5}) = \frac{20}{15}$ . $\newline$ We can simplify this by dividing both numerator and denominator by $5$ , which gives us $\frac{4}{3}$ .
Raise to Power: Raise the simplified fraction to the power of $3$ . Now we have $(\frac{4}{3})^{3}$ . To raise a fraction to a power, we raise both the numerator and the denominator to that power. So, $(\frac{4}{3})^{3} = 4^{3} / 3^{3} = 64 / 27$ .