Powers of Fractions and Negative Exponents

Calculate the numerical value of the following powers. Remember that you must raise both the numerator and the denominator to the indicated 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 foot. 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. Observe the following graph that represents the unit, $\left(\frac{1}{2}\right)^1$ and $\left(\frac{1}{2}\right)^2$.

Based on the previous logic, 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 be able to operate. Transform 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 transforming them into powers with a positive exponent. 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 needing to perform the full calculation. Reason 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. You must leave the result expressed as a single power (you do not need to calculate the final value).

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

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

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

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

Find the missing number in each gap so that the equalities are true. Write the complete equality 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. $\left( \frac{3}{7} \right)^3 \cdot \left( \frac{3}{7} \right)^{\square} = \left( \frac{3}{7} \right)^5$

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

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) $(\frac{x}{y})^{-1} \cdot (\frac{y}{x})^{-1} = 1$
4. d) $\frac{a^3}{b^3} = (\frac{a}{b})^3$

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

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

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

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, just the simplified power.

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

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

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

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

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

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

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

Find the value of the unknown $x$ in each of the following equalities, using the properties of powers with a rational base:

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 \cdot 2^3}{2^{-1}} = 32 $

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

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

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.

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

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

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

We have a special rubber ball. Each time it bounces on 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 feet:

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

Look at the figure of a magic cube. Suppose the edge of a large cube measures $L$.

If we reduce the length of the edge by half ($\frac{1}{2}L$), 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 figure is repeated at different scales. Imagine we design 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 colored 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 up of small unit cubes. Imagine we build giant cubes made up of small cubes with a 1 inch edge.

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

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

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.

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

The storage capacity of computers has grown exponentially.

Knowing 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 how many music files, which take up an average of $4 \cdot 10^6$ bytes each, can fit approximately on a 2 TB hard drive. Express the result in standard form and as an approximate power with base 2 (considering $10^3 \approx 2^{10}$).