Integer Exponents

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) $-2^3$

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

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

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

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

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

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

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) $(-4)^{-2}$

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

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 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 $y$:

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.

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 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 $4 \text{ MB}$.

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?

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:

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.

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

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.

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 $2$.

The radius of a hydrogen atom is approximately $5 \times 10^{-11}$ meters. 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 scientific notation and simplify the expression:

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?