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

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

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

State whether the following statements are True or False. If any are false, write the correct equation.

1. a) $(\frac{a}{b})^{-n} = - (\frac{a}{b})^n$
2. b) $(\frac{2}{3})^2 + (\frac{2}{3})^3 = (\frac{2}{3})^5$
3. c) $(\frac{x}{y})^{-1} \times (\frac{y}{x})^{-1} = 1$
4. d) $\frac{a^3}{b^3} = (\frac{a}{b})^3$

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

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 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) $(\frac{2}{3})^4 \times (\frac{6}{5})^4$

Simplify the following expressions. To do this, you must first break down the bases into prime factors so that all bases are equal, and then apply the properties of powers.

1. a) $(\frac{4}{9})^{-2} \times (\frac{2}{3})^5$
2. b) $(\frac{25}{4})^3 : (\frac{5}{2})^{-2}$

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) $[(\frac{2}{9})^x]^2 = (\frac{2}{9})^{10}$

Work out the following combined operations step by step, respecting the order of operations. Express the final result as an irreducible fraction.

$$ \left( \frac{2}{3} - 1 \right)^{-2} + \left( \frac{1}{2} \right)^3 \times 2^4 - \left( \frac{3}{4} \right)^0 $$

Find the value of the unknown $x$ in each of the following equations, using the properties of powers with rational bases:

1. a) $ \left(\frac{1}{2}\right)^x = 16 $
2. b) $ \left(\frac{3}{5}\right)^{x-1} = \frac{125}{27} $
3. c) $ \frac{2^x \times 2^3}{2^{-1}} = 32 $

Calculate the exact value of the following combined operation with fractions and integer exponents. Remember to respect the order of operations and work with irreducible fractions.

$$ \left( \frac{3}{2} - 2 \right)^{-2} \times \left[ \frac{1}{2} - \left( \frac{1}{4} \right)^{-1} \times \left( \frac{2}{3} \right)^{2} \right] - \left( \frac{3}{4} \right)^0 $$

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.

Look at the figure of a magic cube. Suppose that the edge of a large cube has length $L$.

If we halve the edge length ($\frac{1}{2}L$), how does its volume change?

1. a) Calculate the volume of the original cube ($V_{grande}$) and the reduced cube ($V_{pequeño}$) in terms of $L$.
2. b) Express the ratio $\frac{V_{pequeño}}{V_{grande}}$ as a power with base $\frac{1}{2}$.
3. c) If the large cube has a volume of $512 \text{ cm}^3$, what is the volume of the reduced cube?

In architecture, fractal designs are often used where a shape repeats at different scales. Imagine designing a triangular stained-glass window following this pattern:

We start with an equilateral triangle of area $A$. In step 1, we divide the triangle into 4 equal triangles and remove the central one (the inverted triangle). In step 2, we repeat the process with the 3 remaining triangles, and so on.

1. Write the power with a rational base that represents the fraction of the area that remains coloured after step $n$.

2. If the initial area is $256 \text{ cm}^2$, calculate the remaining area after 4 iterations.

A standard $3 \times 3 \times 3$ Rubik's cube is made of small unit cubes. Imagine that we build giant cubes made of small 1 cm unit cubes.

1. If we have a giant cube whose volume is $8^{4} \text{ cm}^3$, express its edge length as a power with base 2.

2. If we paint the exterior of that giant cube, how many unit cubes will have at least one face painted? Express the calculation using powers and the subtraction of volumes.

Simplify the following algebraic expression using the properties of powers. The final result must not contain negative exponents and must be simplified as much as possible. Be very careful with the signs of the bases and the brackets.

$$ \left[ \frac{(x^2 \times y^{-3})^{-2} \times (x^{-1} \times y^2)^3}{(x^3 \times y^{-4})^{-1}} \right]^{-2} $$

Simplify the following expression by breaking down the bases into prime factors. The result must be expressed as a single power or a product of powers with prime bases.

$$ \frac{27^{-2} \times (-12)^4 \times 9^{-1}}{18^3 \times (-3)^{-5}} $$

Order the following numbers from smallest to largest without calculating their decimal value, using the properties of powers to compare them (hint: try to make the bases or the exponents equal). Justify your answer step by step.

$$ A = 2^{60} \quad ; \quad B = 3^{40} \quad ; \quad C = 5^{30} \quad ; \quad D = (0,5)^{-50} $$

Computer storage capacity has grown exponentially.

Given that:

• 1 Kilobyte (KB) = $2^{10}$ bytes

• 1 Megabyte (MB) = $2^{10}$ KB

• 1 Gigabyte (GB) = $2^{10}$ MB

• 1 Terabyte (TB) = $2^{10}$ GB

Calculate approximately how many music files, averaging $4 \times 10^6$ bytes each, fit on a 2 TB hard drive. Express the result in standard form and as an approximate power with base 2 (taking $10^3 \approx 2^{10}$).