# Which of the following numbers is not equal to

Question:

Which of the following numbers is not equal to $\frac{-8}{27} ?$

(a) $\left(\frac{2}{3}\right)^{-3}$

(b) $-\left(\frac{2}{3}\right)^{3}$

(c) $\left(-\frac{2}{3}\right)^{3}$

(d) $\left(\frac{-2}{3}\right) \times\left(\frac{-2}{3}\right) \times\left(\frac{-2}{3}\right)$

Solution:

(a) $\left(\frac{2}{3}\right)^{-3}$

We can write $\frac{-8}{27}$ as $\frac{-2 \times(-2) \times(-2)}{3 \times 3 \times 3}$. It can be written in the forms given below.

$\frac{-2 \times(-2) \times(-2)}{3 \times 3 \times 3}=-\frac{2 \times 2 \times 2}{3 \times 3 \times 3}$  ---> work out the minuses

$=-\frac{2}{3} \times \frac{2}{3} \times \frac{2}{3}$

$=-\left(\frac{2}{3}\right)^{3}$

Hence, option (b) is equal to $\frac{-8}{27}$.

We can also write:

$\frac{-2 \times(-2) \times(-2)}{3 \times 3 \times 3}=\left(-\frac{2}{3}\right) \times\left(-\frac{2}{3}\right) \times\left(-\frac{2}{3}\right)$

$=-\left(\frac{2}{3}\right)^{3}$

Hence, option (c) is also equal to $\frac{-8}{27}$.

We can also write:

$\frac{-2 \times(-2) \times(-2)}{3 \times 3 \times 3}=\left(-\frac{2}{3}\right) \times\left(-\frac{2}{3}\right) \times\left(-\frac{2}{3}\right)$

Hence, option (d) is also equal to $-\frac{8}{27}$.

This leaves out option (a) as the one not equal to $-\frac{8}{27}$.