Powers of rational numbers

We have a small square tile for a mosaic. The side of the tile measures $\frac{2}{5}$ of a metre. Calculate the area occupied by this tile, expressing the result as a fraction.

Use the following visual diagram to help you:

Sometimes we confuse the size of fractions when we raise them to a power. Look at the following graph showing the unit, $\left(\frac{1}{2}\right)^1$ and $\left(\frac{1}{2}\right)^2$.

Based on the logic above, order the following powers from smallest to largest:

$$ \left( \frac{1}{3} \right)^2, \quad \left( \frac{1}{3} \right)^1, \quad \left( \frac{1}{3} \right)^3 $$

Calculate the numerical value of the following powers. Remember to raise both the numerator and the denominator to the given exponent. Express the result as an irreducible fraction.

1. $\left( \frac{2}{3} \right)^3$

2. $\left( \frac{1}{5} \right)^2$

3. $\left( \frac{3}{2} \right)^4$

4. $\left( \frac{10}{7} \right)^0$

In mathematics, we sometimes need to change the sign of the exponent to perform operations. Rewrite the following powers so that they have a positive exponent, inverting the base when necessary. You do not need to calculate the final result, just express the transformed power.

1. $\left( \frac{4}{5} \right)^{-2}$

2. $\left( \frac{1}{3} \right)^{-4}$

3. $5^{-3}$

4. $\left( -\frac{2}{7} \right)^{-3}$

Calculate the value of the following powers by first converting them into powers with positive exponents. Remember that the result must be an irreducible fraction.

1. a) $(\frac{3}{5})^{-2}$
2. b) $(\frac{-2}{3})^{-3}$
3. c) $-( \frac{1}{4} )^{-2}$
4. d) $(\frac{10}{3})^{-1}$

Determine the sign (positive or negative) of the result of the following powers without carrying out the full calculation. Justify your answer based on whether the base is negative and whether the exponent is even or odd.

1. $\left( -\frac{1}{2} \right)^4$

2. $\left( -\frac{3}{4} \right)^3$

3. $\left( \frac{5}{2} \right)^{-2}$

4. $\left( -\frac{2}{3} \right)^{-5}$

Calculate the following powers that have the same exponent but different bases. Remember that you can multiply the bases first and raise the result to the common exponent. Express the result as a simplified fraction.

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

2. $\left( \frac{1}{3} \right)^2 \times 6^2$

3. $\left( \frac{3}{2} \right)^4 : \left( \frac{1}{2} \right)^4$

Use the properties of powers to simplify the following expressions. Leave your answer expressed as a single power (you do not need to calculate the final value).

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

2. $\left( \frac{2}{5} \right)^7 : \left( \frac{2}{5} \right)^4$

3. $\left( \frac{1}{2} \right)^3 \times \left( \frac{1}{2} \right)$

4. $\left[ \left( \frac{5}{3} \right)^2 \right]^4$

Find the missing number in each box to make the equations true. Write the complete equation in your notebook.

1. $\left( \frac{2}{3} \right)^{\square} = \frac{8}{27}$

2. $\left( \frac{5}{2} \right)^{\square} = 1$

3. $\left( \frac{1}{4} \right)^{-2} = \square$

4. $\left( \frac{3}{7} \right)^3 \times \left( \frac{3}{7} \right)^{\square} = \left( \frac{3}{7} \right)^5$

Find the value of $x$ in each of the following equations to make them true. Think about the properties of powers with the same base.

1. a) $(\frac{1}{2})^x = 32$
2. b) $(\frac{3}{5})^4 \times (\frac{3}{5})^x = (\frac{3}{5})^{-1}$
3. c) $[(\frac{2}{9})^x]^2 = (\frac{2}{9})^{10}$