Powers of Rational Numbers

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

We have a small square tile for a mosaic. The side of the tile measures $\frac{2}{5}$ of a meter. 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:

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

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

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

Use the properties of powers to simplify the following expressions and write the result as a single power with a rational base. You do not need to calculate the final value, only the simplified power.

1. a) $(\frac{2}{7})^3 \times (\frac{2}{7})^5 : (\frac{2}{7})^{-2}$
2. b) $[(\frac{1}{5})^2]^{-3} \times (\frac{1}{5})^4$
3. c) $(\frac{2}{3})^4 \times (\frac{6}{5})^4$

We have a special rubber ball. Each time it bounces off the ground, it reaches a height that is $\frac{2}{3}$ of the previous height.

If we drop the ball from a height of 2 meters:

1. a) Express as a power the height it reaches after the fourth bounce.
2. b) Is that height greater or less than 0.3 meters? Justify your answer using the power calculation.