Powers with integer indices

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

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

$$A = (-2)^{-4} \times 4^2 - (-3)^0$$

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:

$$A = 16 \times 16 - (-1) = 256 + 1 = 257"$$

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.

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:

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

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

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

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:

$$V = (5 \times 10^{-11})^3$$

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

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

$$\frac{(x^2 \times x^3)^{-2}}{x^{-8}}$$

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

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

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:

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

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

$$\frac{6^a \times 10^b}{15^c}$$

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.

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:

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

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 $3^4 \times 3^t$.

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:

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

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.