Exercises: Powers with integer exponents

Apply the laws of indices to simplify the following expression and leave the result as a single power with a positive base:

Calculate the numerical value of the following powers with negative exponents, expressing the final result as an irreducible fraction:

1. a) $5^{-2}$

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

1. c) $(-4)^{-2}$

It is very common to confuse the effect of brackets when the base is negative. Calculate the following values and explain the difference between them:

1. a) $(-2)^4$

1. b) $-2^4$

1. c) $(-2)^3$

1. d) $-2^3$

Simplify the following expression using the power of a power and product of powers laws. Express the result as a single power:

In computing, units of measurement grow in powers of base $2$. Knowing that $1 \text{ KB (Kilobyte)}$ is $2^{10} \text{ bytes}$ and that $1 \text{ MB (Megabyte)}$ is $2^{10} \text{ KB}$, express as a single power of base $2$ how many bytes there are in an image file that takes up $4 \text{ MB}$.

Reduce the following expression to a single power of base $x$ and then write it with a positive exponent:

A biologist observes a population of bacteria that triples every hour. If initially there were $3^2$ bacteria, express using a power of base $3$:

1. a) How many bacteria will there be after $5$ hours?
2. b) How many bacteria will there be after $n$ hours?

Calculate the value of the following combined expressions, respecting the order of operations and the laws of indices:

1. a) $2^3 + 2^2 + 2^1 + 2^0$

1. b) $10^3 \times 10^{-2} + 5 \times 10^0$

1. c) $(3-1)^2 \times 3^{-1}$

In a 5th Year exam, Aoife had to simplify the following numerical expression:

Aoife wrote on her answer sheet:

"Since the base is negative and the exponent is also negative, the two minus signs cancel each other out, so that $(-2)^{-4} = 2^4 = 16$. On the other hand, since any number raised to 0 is 1, but it has a minus sign in front inside the parentheses, $(-3)^0 = -1$. Therefore:

1. a) Identify and explain in detail the two conceptual errors made by Aoife, stating which property of powers or sign rule she violated in each.
2. b) Correct the solution by calculating step by step the exact value of the expression as an irreducible fraction or an integer.

Consider the following numerical expression with integer exponents:

1. a) Estimate reasonably without calculating the exact value:

Observe the factors in the numerator and denominator and interpret the effect of the powers with negative exponents when moving them across the fraction. Reason whether the final value of $E$ will be:

1. Positive or negative.

2. Greater than $1$ or less than $1$.

1. b) Calculate and check:

Decompose all bases into their prime factors ($2$ and $5$), apply the properties of powers to simplify the expression step by step until you obtain a single integer or irreducible fraction, and explicitly verify whether the numerical result confirms the predictions you made in the previous section.

To calculate the storage capacity of a new cloud server, an engineer needs to simplify the following numerical expression. Use the laws of indices to reduce it to a single power with a prime base or a product of powers with positive exponents:

A cybersecurity firm uses cryptographic keys whose length grows exponentially. If a level 1 key has a complexity of $(x^2 \times y^{-3})^2$ and is multiplied by a level 2 key with complexity $(x \times y^2)^{-4}$, simplify the resulting expression to find the total complexity of the system in terms of $x$ and $y$:

In a biotechnology laboratory, a bacterial population triples every hour. If initially there were $3^4$ bacteria, the expression representing the number of bacteria after $t$ hours is $3^4 \times 3^t$.

1. a) How many bacteria will there be after one day ($t=24$)? Express the result as a power.

1. b) If due to an antibiotic the population decreases by dividing by $9^5$, simplify the resulting expression:

Work out and simplify the following expression, leaving the result as an irreducible fraction without negative exponents:

A communications satellite sends data packets of size $2^{15}$ bytes. If the satellite must send a total file occupying $16^5$ bytes, calculate how many packets it will need to transmit. Express both the initial data and the final result as powers of base 2.

State whether the following equalities are true or false, reasoning your answer based on the laws of indices and the rule of signs:

1. a) $(-2)^4 = -2^4$

1. b) $\frac{a^{-3}}{b^{-3}} = \left( \frac{b}{a} \right)^3$

1. c) $(x+y)^2 = x^2 + y^2$

1. d) $(-5)^{-2} = \frac{1}{25}$

The radius of a hydrogen atom is approximately $5 \times 10^{-11}$ metres. If we wanted to represent the volume of a cubic space having exactly that radius as its edge, we would have to cube that measurement ($V = l^3$).

Calculate the resulting volume in standard form and simplify the expression:

Order the following values from smallest to largest by first transforming them to have the same base:

We want to construct a fraction formed exclusively by powers of non-prime bases of the form:

where $a$, $b$, and $c$ are integers (positive, negative, or zero). The goal is that, after applying the properties of powers and simplifying as much as possible, the exact result is the irreducible fraction $\frac{20}{27}$.

1. a) Decompose the composite bases ($6$, $10$, $15$) and the numbers of the target fraction ($20$, $27$) into prime factors.
2. b) Set up the system of linear equations that the exponents $a$, $b$, and $c$ must satisfy by equating the powers of each prime base on both sides.
3. c) Solve the system to determine the integer values of $a$, $b$, and $c$. Identify which of them turns out to be strictly negative and explain what practical meaning this negative exponent has regarding the actual position of that factor in the fraction.

A telecommunications company monitors monthly data consumption in a region of Castilla-La Mancha. Taking the year 2024 as the time reference ($k = 0$), the increase in remote work causes average monthly traffic to double exactly every 3 years (that is, every three-year period it is multiplied by a factor of $2$). In 2024, the recorded traffic was $128\text{ TB}$ ($128 = 2^7\text{ TB}$).

1. a) If $k \in \mathbb{Z}$ represents the number of three-year periods that have elapsed since 2024 (where $k>0$ indicates three-year periods in the future and $k<0$ three-year periods in the past), write the formula for monthly traffic $T(k)$ in terabytes as a single base-$2$ power.
2. b) What calendar year in the past corresponds to the value $k = -3$? Calculate the data traffic for that year by applying the properties of negative exponents and justify why the minus sign models going back in time (successively dividing by $2$ instead of multiplying).
3. c) For the year 2036, it is planned to install a central node with a capacity of $2^{15}\text{ TB}$. Use the quotient of powers of the same base to calculate how many times greater this capacity will be compared to the traffic recorded in the year corresponding to $k = -3$.

Simplify the following algebraic expression until you obtain a single power or a product of powers with prime bases, eliminating all negative exponents and without using a calculator:

To solve it, first express all the bases as products of prime factors ($2$ and $3$) and carefully apply the laws of indices.

In the study of economic models, expressions sometimes appear that require taking out a common factor from powers with variable exponents. Simplify the following fraction as much as possible to make it easier to program in a spreadsheet:

A cybersecurity company is analysing the spread of a computer virus. It has been observed that the number of infected files triples every hour. However, every $4$ hours, an antivirus software manages to eliminate $80 \%$ of the infections existing at that moment.

If initially there is only $1$ infected file, express using an operation with powers the number of affected files after $12$ hours, right after the third antivirus action. Could you write the final result in standard form?

Work out and simplify the following expression with variables, leaving the final result without negative exponents:

An investor analyses two financial products. In the first, their capital is multiplied by $1.02$ every month (compound interest). In the second, due to high volatility, the capital is multiplied by $1.05$ one month, but decreases by being multiplied by $0.98$ the following month, repeating this two-month cycle indefinitely.

If the investor deposits an amount $C$ for $2$ years ($24$ months), express using powers with integer exponents the final capital obtained in each option. Which of the two options is more profitable after those $24$ months? Justify your answer by comparing the bases of the resulting powers, once they are written with the same exponent.

Calculate the value of the following numerical expression by expressing the bases in prime factors and simplifying as much as possible. The result must be an integer or an irreducible fraction:

In the world of computing, storage capacities grow exponentially. Suppose a server doubles its storage capacity every $18$ months. If in the year $2020$ it had a capacity of $2^{10}$ Terabytes (TB):

1. a) Express using a power of base $2$ what its capacity will be in the year $2035$.
2. b) If each TB is equivalent to $2^{40}$ bytes, express the total capacity in the year $2035$ in bytes using a single power of base $2$.

Prove, applying the properties of powers, that the following equality is true for any value of $x$ and $y$ (provided the bases are non-zero):

Indicate step by step which property you are applying at each stage of the simplification.

To simplify the following algebraic expression (with $a, b \ne 0$):

two 1st Year students propose different approaches:

• Aoife's method: First, she flips the entire fraction to change the outer exponent $-2$ to $+2$; then she moves each factor with a negative exponent to the opposite side (numerator or denominator) so that all exponents become positive; finally, she factors $18$ and $12$ into primes and simplifies.

• Seán's method: He keeps the original fraction and first operates inside the brackets: he factors $18$ and $12$ into primes, distributes the power of the product in the denominator, groups powers with the same base by subtracting exponents ($a^m / a^n = a^{m-n}$ operating with negative integers) and, only at the end, applies the power of a power with the outer exponent $-2$.

1. a) Carry out the simplification step by step following both methods and verify that both lead to the same irreducible expression with positive exponents.
2. b) Critically compare both strategies: which one minimizes the risk of making mistakes when subtracting negative numbers? Point out precisely in which specific algebraic step it is easiest to make a mistake in each of the methods.

Find all pairs of integers $(m, n) \in \mathbb{Z} \times \mathbb{Z}$ that satisfy the following equation:

To solve the problem without trial and error, develop the following sections with justification:

1. a) Sign and parity: Note that the right-hand side is negative. Knowing that $6^{n-m} > 0$ and $9^{-n} > 0$ for any integers $m$ and $n$, deduce the sign that the power $(-2)^m$ must have and prove what parity condition (even or odd) the integer $m$ must necessarily satisfy.
2. b) Decomposition into prime bases: Rewrite both sides of the equation by decomposing all bases into their prime factors ($2$ and $3$) and apply the properties of operations with powers to reduce the left-hand side to the form $- (2^P \times 3^Q)$, where $P$ and $Q$ are algebraic expressions in terms of $m$ and $n$.
3. c) Solution and uniqueness: Using the uniqueness of prime factorization in rational numbers, set up and solve the system of equations with unknowns $m$ and $n$. Check that the obtained solution satisfies the parity condition found in (a) and justify why there can be no other integer solution.