Exercises: Operations with powers

Write the following repeated multiplications as a power and then calculate their numerical value. Remember that the base is the number being repeated and the exponent indicates how many times it appears.

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

Use the properties of multiplying powers with the same base to simplify these expressions and leave them as a single power. You do not need to calculate the final result, just express the resulting power.

1. a) $5^3 \cdot 5^4$
2. b) $(-3)^2 \cdot (-3)^5$
3. c) $x^2 \cdot x^6 \cdot x$
4. d) $10^3 \cdot 10^2 \cdot 10^4$

Apply the quotient property (division) of powers with the same base to simplify the following expressions to a single power. Remember to subtract the exponents.

1. a) $\frac{7^8}{7^3}$
2. b) $2^{12} : 2^{10}$
3. c) $\frac{(-4)^5}{(-4)^2}$
4. d) $\frac{a^6}{a}$

We have a classic Rubik's cube. We know that on each edge (side) there are 3 small pieces.

Express using a power how many small pieces make up the whole cube and calculate the value.

Look at the drawing to help you:

Calculate the value of the following powers taking into account the properties of zero exponents and powers of base 1. Be careful with the signs.

1. a) $5^0$
2. b) $23^1$
3. c) $1^{15}$
4. d) $(-8)^0$
5. e) $123^0 + 1^{20}$

Simplify the following expressions using the power of a power property. Leave your answer as a single power.

1. a) $(3^2)^4$
2. b) $(x^3)^5$
3. c) $((-2)^2)^3$
4. d) $((a^2)^3)^4$

In these cases, the bases are different but the exponents are equal. Apply the corresponding properties to group the bases and express the result as a single power.

1. a) $2^3 \cdot 5^3$
2. b) $12^4 : 4^4$
3. c) $(-2)^5 \cdot 3^5$
4. d) $\frac{20^2}{5^2}$

Fully simplify the following expressions by combining the properties of powers you have learned (product and quotient of equal bases). Express the result as a single power.

1. a) $\frac{2^5 \cdot 2^3}{2^4}$
2. b) $\frac{x^7}{x^2 \cdot x^3}$
3. c) $(3^2 \cdot 3^4) : 3^5$
4. d) $\frac{10^6 \cdot 10^2}{10^8}$

In a 3rd year ESO exam, Clara must simplify the operation $2^3 \cdot 2^4$. She writes the following working out in her notebook:

$$2^3 \cdot 2^4 = (2 \cdot 2)^{3+4} = 4^7$$

1. a) Identify and describe the two distinct conceptual errors Clara made in her procedure.
2. b) Explain, using the basic meaning of a power as repeated multiplication (without just citing the formula from memory), why when multiplying powers with the same base, the exponents are added but the base remains unchanged.
3. c) Correct the exercise by writing the proper solution as a single power with base 2 and calculate its final numerical value.

Design three different arithmetic operations whose final result is exactly the power $3^8$, strictly fulfilling the indicated condition in each case:

1. a) A product of two powers with base 3 where neither exponent is an even number.
2. b) A division between two powers that have different bases and the same exponent.
3. c) A power of another power where one of the exponents involved is a negative integer.

Justify in each section the property of powers that guarantees that your proposal results in exactly $3^8$.

Work out the result of the following expression, applying the properties of powers before calculating the final value. Note that all bases are the same negative number:

$$ \frac{(-3)^5 \cdot (-3)^2 \cdot (-3)}{(-3)^4} $$

Remember that when a number does not have a visible exponent, it is 1.

Simplify the following operation and express the result as a single power of base 2. To do this, you will first need to break down into prime factors any bases that are not prime numbers:

$$ 16 \cdot 2^5 \cdot 4^3 $$

We have a square whose side measures $2^4$ centimeters.

1. Express the perimeter of the square as a single power of base 2 (you do not need to calculate the final number, just the power).

2. Express the area of the square as a single power of base 2.

Calculate and simplify the following expression using the properties of powers with rational bases. Express the result as an irreducible fraction with a positive exponent:

$$ \left[ \left( \frac{2}{5} \right)^{-2} \cdot \left( \frac{2}{5} \right)^5 \right] : \left( \frac{5}{2} \right)^2 $$

An external hard drive has a storage capacity of 1 TB (Terabyte). Knowing that the approximate equivalences in the binary system are:

• $1 \text{ TB} = 2^{10} \text{ GB}$

• $1 \text{ GB} = 2^{10} \text{ MB}$

If a high-quality photograph takes up $2^4$ MB, how many photographs fit in total on the hard drive? Express the result as a power of base 2.

Simplify the following algebraic expression using the properties of powers. The final result must not have negative exponents:

$$ \frac{a^3 \cdot b^{-4} \cdot a^{-2}}{a^5 \cdot b^{-6}} $$

Find the value of the unknown $x$ in each of the following equations, using the properties of powers to equate the exponents:

1. a) $5^x \cdot 5^3 = 5^9$
2. b) $(3^4)^x = 3^{12}$
3. c) $\frac{7^8}{7^x} = 7^2$

Simplify the following numerical fraction by first factorising the bases into prime factors and then applying the properties of powers:

$$ \frac{15^2 \cdot 4^2}{6^3 \cdot 10^2} $$

To simplify and calculate the numerical value of the expression $\frac{12^4 \cdot 5^4}{30^3}$, Sergio and Alba debate which strategy is more convenient:

• Sergio proposes first breaking down all the bases into their prime factors ($12 = 2^2 \cdot 3$, $30 = 2 \cdot 3 \cdot 5$) and using the properties of the product and quotient of powers with the same base.

• Alba proposes first applying the property of powers with the same exponent in the numerator to group it as $(12 \cdot 5)^4 = 60^4$, and then expressing 60 as $2 \cdot 30$ to work with the base 30 of the denominator.

1. a) Solve the expression following Sergio's proposal step by step.
2. b) Solve the expression applying the strategy proposed by Alba.
3. c) Compare both paths: which of the two methods do you consider more efficient for this specific case? Justify your choice by analyzing the volume of calculations and the risk of making mistakes in each one.

The Ministerio para la Transición Ecológica y el Reto Demográfico (MITECO) coordinates reforestation efforts in a region of the Sierra de Gata (Extremadura). To organize the repopulation of native trees, the land has been divided into $3^3$ main plots. Within each plot, $9^2$ sectors have been set up, and in each of these sectors, exactly 27 holm oak saplings have been planted.

1. a) Set up a single combined operation with powers that represents the total number of holm oaks planted in the region.
2. b) Break down the bases into prime factors and apply the properties of powers to express the total number of holm oaks as a single power of base 3. How many trees have been planted in total?
3. c) During the first few weeks, each sapling needs to receive $3^{-2}$ liters of water per hour through a localized irrigation system. Express as a single power of base 3 how many liters of water the entire plantation consumes in one hour.

Simplify the following expression to obtain a single power with a positive prime base. To do this, you will need to apply the properties of powers, factorise the bases into prime factors, and be very careful with signs and negative exponents.

$$ \frac{(-2)^4 \cdot (2^3)^{-2} \cdot 16}{\left(\frac{1}{2}\right)^3 \cdot (-8)^{-2}} $$

Remember that factorising numbers like 16 and 8 into base 2 is the key first step.

Observe the following sequence of figures generated by a fractal process.

1. Start with a solid square of side 1 metre (Iteration 0).

2. Divide the square into 4 equal squares and remove the top right one (Iteration 1).

3. Repeat the process with each of the 3 remaining squares (Iteration 2).

Calculate the total area (the sum of the areas of the black squares) in Iteration 5. Express the result as a single power.

Find the value of the unknown $x$ in the following exponential equation, using the properties of powers to collect terms. All bases can be expressed as powers of the same prime number.

$$ 27^2 \cdot 9^{-x} = \frac{1}{3^x \cdot 81} $$

A cloud data storage server has a total capacity of 16 Petabytes (PB). Knowing that computing equivalences are powers of 2 (not 10) according to the following table:

• $1 \text{ KB} = 2^{10} \text{ bytes}$

• $1 \text{ MB} = 2^{10} \text{ KB}$

• $1 \text{ GB} = 2^{10} \text{ MB}$

• $1 \text{ TB} = 2^{10} \text{ GB}$

• $1 \text{ PB} = 2^{10} \text{ TB}$

If a high-resolution photograph file takes up $2^{24}$ bytes, how many photographs fit in total on the server? Express the result as a power of base 2.

Without using a calculator to find the final value, order the following numbers from smallest to largest. To do this, you will need to find a common exponent or a common base to allow you to compare them mentally.

$$ A = 2^{45} \quad ; \quad B = 3^{36} \quad ; \quad C = 5^{18} $$

Help: Try expressing all exponents as multiples of the same number (find the highest common factor of the exponents).

Simplify the following algebraic expression using the properties of powers. The final result must not contain negative exponents or brackets.

$$ \left( \frac{a^{-2} \cdot b^3}{a \cdot b^{-1}} \right)^{-3} : \left( \frac{a^3}{b^2} \right)^2 $$

A giant Rubik's cube is made of small unit cubes. We know that the total volume of the large cube is $27^4$ cubic units.

1. Express the length of the edge of the large cube as a power of base 3.

2. If we paint the outer face of each unit cube red, what is the total area painted red on the entire giant cube? Express the result in the form $2 \cdot 3^n$.

(Remember that the large cube has 6 faces and the area of a face is side $\cdot$ side).

Work out the exact value of the following combined operation, applying the properties of powers first and converting to standard form before working it out.

$$ \frac{ 3,5 \cdot 10^{-5} + 150 \cdot 10^{-7} }{ 0,02 \cdot 10^4 \cdot (10^{-2})^3 } $$

Express the final result in standard form.

The Instituto Nacional de Estadística (INE) keeps backups of census data in two independent server centers, referred to as Node A and Node B:

• Node A stores $8^7$ data blocks.

• Node B stores $4^{11}$ data blocks.

1. a) Before performing calculations in writing, make a reasoned intuitive estimate: which of the two nodes do you think contains more data blocks? Explain which element carried more weight in your first impression (the value of the base or the magnitude of the exponent?).
2. b) Rigorously verify your estimate: express the amount of blocks in each node as a power of base 2 by applying prime factorization and the power of a power property.
3. c) Calculate the quotient between the node with the greater capacity and the one with the smaller capacity. How many times more information does one hold compared to the other?
4. d) In light of the result, explain why when comparing exponential quantities, visual intuition based solely on the apparent size of the base often leads to errors.

A cybersecurity team in León generates encrypted access codes. A master code $K$ is defined by the following expression:

$$K = \frac{15^a \cdot 18^b}{10^c \cdot 12^d}$$

where $a$, $b$, $c$, and $d$ are positive integers.

When simplifying the expression by decomposing the bases into prime factors and applying the properties of powers, the analysts observe that bases 2 and 5 cancel out completely (their resulting exponents are equal to 0), so that the final result simplifies to $K = 3^6$.

1. a) Decompose the composite bases ($15$, $18$, $10$, and $12$) into prime factors and write the expression by grouping the exponents of bases 2, 3, and 5 in terms of the unknowns $a$, $b$, $c$, and $d$.
2. b) Write the equations that the exponents of bases 2 and 5 must satisfy so that both cancel out in the final result.
3. c) If the system configuration sets $c = 1$ and $d = 1$, deduce with reasoning the exact numerical values that exponents $a$ and $b$ must have, and verify step by step that the value obtained for $K$ matches $3^6$.