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

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

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

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:

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

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:

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

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

Cian writes the following solution on the board for an exercise proposed by his teacher:

$$\left(-\frac{3}{5}\right)^{-2} = -\left(\frac{5}{3}\right)^2 = -\frac{25}{9}$$

Defending his answer, he argues: "The minus sign in the exponent turns the entire power into a negative number, and then I just have to flip the fraction and square it."

1. a) Identify the two conceptual errors made by Cian. Clearly explain the difference between what the minus sign in the base does and what the minus sign in the exponent does.
2. b) Correct the exercise by calculating the exact value as an irreducible fraction. Finally, reason what sign the result would have had if the initial exponent had been $-3$ instead of $-2$.

Without performing detailed written calculations at first, look closely at these three powers:

$$A = \left(\frac{2}{3}\right)^4, \quad B = \left(\frac{2}{3}\right)^{-3}, \quad C = \left(\frac{3}{2}\right)^{-3}$$

1. a) Make a reasoned prior estimation: determine which of these powers will yield a result less than $1$ and which will yield a result greater than $1$. Furthermore, anticipate whether two of them will give exactly the same final value. Justify your prediction using the properties of fractions and negative exponents.
2. b) Calculate the exact numerical value of $A$, $B$, and $C$ as irreducible fractions and compare the results with your initial estimation to check whether your reasoning was correct.

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) $(\frac{10}{3})^{-1}$

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 metres:

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 metres? Justify your answer using the power calculation.

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

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?

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

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

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) $\frac{a^3}{b^3} = (\frac{a}{b})^3$

In a report by the Office of Public Works (OPW) on drought monitoring in the Shannon catchment, technicians estimate that the flow rate of a tributary of the Shannon decreases steadily during dry periods: each month that passes without rain, the flow rate becomes $\frac{2}{3}$ of the flow rate of the previous month. Hydrologists model this evolution with the expression $C(t) = C_0 \times \left(\frac{2}{3}\right)^t$, where $C_0$ is the current flow rate (in $\text{m}^3/\text{s}$) and $t$ represents the elapsed time in months ($t > 0$ future months, $t < 0$ past months).

1. a) A technician states that to calculate the flow rate the tributary had 3 months ago, the factor $\left(\frac{2}{3}\right)^{-3}$ must be applied. Calculate this factor as an irreducible fraction and explain the physical and mathematical meaning of the fact that, by inverting the fraction, the factor obtained is strictly greater than $1$.
2. b) If the flow rate measured at the gauging station today is $C_0 = 16\text{ m}^3/\text{s}$, what was the estimated flow rate 2 months ago ($t = -2$)? And what will it be in 2 months ($t = 2$)? Express both results as irreducible fractions.

Find an irreducible fraction $\frac{a}{b}$ and an integer $n$ that simultaneously satisfy the following three conditions:

1) $\left(\frac{a}{b}\right)^n = \frac{81}{16}$

2) The exponent $n$ is a negative integer strictly less than $-1$.

3) The base fraction satisfies $0 < \frac{a}{b} < 1$.

1. a) Deduce step by step all possible pairs of values of $\frac{a}{b}$ and $n$, justifying how you use the prime factorisation of the numerator and denominator.
2. b) If it were not mandatory for the fraction to be irreducible, could you find another different positive base that would give the same result with another negative integer exponent? And what would happen if we allowed the base $\frac{a}{b}$ to be negative? Analyse and justify the possible additional solutions that arise in both cases.

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.

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

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.

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

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.

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.

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

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 $

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.

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 $10^3 \approx 2^{10}$).

To simplify the following expression with fractions and integer exponents:

$$E = \frac{\left[\left(-\frac{3}{4}\right)^{-2}\right]^3 \times \left(\frac{4}{3}\right)^{-5}}{\left(\frac{9}{16}\right)^{-2}}$$

two Year 10 students propose two different strategies:

• Aoife's strategy: She works first with signs and inversions; she converts each power to a positive exponent by inverting the individual fractions, decomposes all bases into their prime factors ($2$ and $3$), and simplifies the resulting large fraction using the properties of powers with natural exponents.

• Cian's strategy: He maintains the structure of rational bases by unifying everything into powers with base $\frac{3}{4}$. To do this, he uses that $\frac{4}{3} = \left(\frac{3}{4}\right)^{-1}$, that $\frac{9}{16} = \left(\frac{3}{4}\right)^2$, analyses the overall sign by observing the even and odd exponents, and algebraically operates on the integer exponents using standard properties before calculating any large numbers.

1. a) Solve the complete expression following Aoife's method.
2. b) Solve the complete expression following Cian's method.
3. c) Compare both approaches: which one minimises the number of calculations and reduces the likelihood of making mistakes with multi-level complex fractions? Justify which of the two methods you would recommend to a classmate.

We want to construct an expression with powers of fractions of the form:

$$M = \frac{\left(\frac{x}{y}\right)^p \times \left(\frac{y}{x}\right)^q}{\left(\frac{x}{y}\right)^r}$$

where $x, y \neq 0$ and the exponents $p, q, r$ are integers. The expression must simultaneously satisfy the following three conditions:

1) When simplified using the properties of powers, the final result must be exactly $\left(\frac{x}{y}\right)^{-3}$.

2) At least two of the three exponents ($p, q, r$) must be strictly negative integers.

3) The exponent $q$ cannot be zero ($q \neq 0$).

1. a) Express condition (1) as an algebraic equation relating the exponents $p$, $q$, and $r$, after first transforming the base $\frac{y}{x}$ into a power with base $\frac{x}{y}$.
2. b) Find a triplet of integers $(p, q, r)$ that satisfies all the required conditions and verify the validity of your solution.
3. c) A classmate claims that it is impossible to find a valid triplet if a fourth condition is added: $p + q + r = 1$. Prove with reasoning whether your classmate's claim is true or false by setting up the corresponding system.