Work out the following without a calculator: a) \( \left(\frac{1}{3}\right)^{2} \) b) \( \left(\frac{2}{5}\right)^{3} \) d) \( \left(\frac{3}{4}\right)^{2} \) e) \( \sqrt{\frac{4}{25}} \) i) \( \left(-\frac{1}{2}\right)^{3} \) g) \( \left(\frac{2}{3}\right)^{3} \) (i) \( 1\left(-\frac{2}{7}\right)^{2} \) k) \( : \sqrt[3]{-\frac{1}{27}} \)

Let's solve each of the expressions step by step. ### a) \( \left(\frac{1}{3}\right)^{2} \) To calculate this, we square the numerator and the denominator: \[ \left(\frac{1}{3}\right)^{2} = \frac{1^{2}}{3^{2}} = \frac{1}{9} \] ### b) \( \left(\frac{2}{5}\right)^{3} \) Here, we cube the numerator and the denominator: \[ \left(\frac{2}{5}\right)^{3} = \frac{2^{3}}{5^{3}} = \frac{8}{125} \] ### d) \( \left(\frac{3}{4}\right)^{2} \) Again, we square the numerator and the denominator: \[ \left(\frac{3}{4}\right)^{2} = \frac{3^{2}}{4^{2}} = \frac{9}{16} \] ### e) \( \sqrt{\frac{4}{25}} \) To find the square root of a fraction, we take the square root of the numerator and the denominator: \[ \sqrt{\frac{4}{25}} = \frac{\sqrt{4}}{\sqrt{25}} = \frac{2}{5} \] ### i) \( \left(-\frac{1}{2}\right)^{3} \) We cube the numerator and the denominator, keeping the negative sign: \[ \left(-\frac{1}{2}\right)^{3} = \frac{(-1)^{3}}{2^{3}} = \frac{-1}{8} \] ### g) \( \left(\frac{2}{3}\right)^{3} \) Cubing the numerator and the denominator: \[ \left(\frac{2}{3}\right)^{3} = \frac{2^{3}}{3^{3}} = \frac{8}{27} \] ### (i) \( 1\left(-\frac{2}{7}\right)^{2} \) First, we square the fraction: \[ \left(-\frac{2}{7}\right)^{2} = \frac{(-2)^{2}}{7^{2}} = \frac{4}{49} \] Then, multiplying by 1 does not change the value: \[ 1 \cdot \frac{4}{49} = \frac{4}{49} \] ### k) \( \sqrt[3]{-\frac{1}{27}} \) The cube root of a negative fraction can be calculated as follows: \[ \sqrt[3]{-\frac{1}{27}} = \frac{\sqrt[3]{-1}}{\sqrt[3]{27}} = \frac{-1}{3} \] ### Summary of Results - a) \( \frac{1}{9} \) - b) \( \frac{8}{125} \) - d) \( \frac{9}{16} \) - e) \( \frac{2}{5} \) - i) \( \frac{-1}{8} \) - g) \( \frac{8}{27} \) - (i) \( \frac{4}{49} \) - k) \( \frac{-1}{3} \)