Powers with 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) $10^3 \cdot 10^{-2} + 5 \cdot 10^0$

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

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

$$\frac{3^5 \cdot 3^2}{3^4}$$

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

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

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:

$$\frac{\left( \frac{2}{3} \right)^{-2} \cdot \left( \frac{4}{9} \right)^{3}}{\left( \frac{3}{2} \right)^{4} \cdot 2^{-1}}$$

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:

$$\frac{12^4 \cdot 18^{-2} \cdot 2^5}{32 \cdot 3^{-3}}$$

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:

$$\frac{15^6 \cdot 12^{-4} \cdot 80^3}{20^4 \cdot 27^2 \cdot 3^{-5}}$$

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 $3^4 \cdot 3^t$.

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:

$$\frac{3^4 \cdot 3^{24}}{9^5}$$

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.