Exercises: Powers with integer exponents

Calculate the numerical value of the following powers with negative exponents, expressing the result as an irreducible fraction or an integer: $3^{-2}$, $(-2)^{-3}$, $5^{-1}$ and $(-1)^{-4}$.

Express the following powers with fractional bases using a positive exponent and then calculate their exact value: $\left(\frac{2}{5}\right)^{-3}$ and $\left(-\frac{1}{4}\right)^{-2}$.

Apply the laws of multiplication for powers of the same base to express the result as a single power: $7^4 \cdot 7^{-6} \cdot 7^5$.

Simplify the following expression by evaluating the operations in the numerator and denominator, and express the final result as a single power: $\frac{5^{-3} \cdot 5^6}{5^2}$.

Reduce the following expression to a single power by applying the power of a power rule: $\left((-2)^3\right)^{-4}$.

Use the product law for powers with the same exponent to express the following calculation as a single power: $4^{-3} \cdot 2^{-3} \cdot 5^{-3}$.

Simplify the following fraction by expressing both the numerator and denominator as powers of the same prime base, and give the result as a single power: $\frac{8^4}{4^5}$.

Perform the necessary operations to express the following product as a single power of base $3$: $9^2 \cdot 27^{-1} \cdot 3^4$.

In a 4th-year ESO exam, Sonia makes three very common mistakes when working with powers with integer exponents:

1) She writes that $(-3)^{-2} = -\frac{1}{9}$, reasoning that "the minus sign in the exponent turns the final result negative".

2) She writes that $5^{-1} = -5$, stating that "raising to $-1$ is the same as changing the sign of the number".

3) She writes that $\left(-\frac{2}{5}\right)^{-3} = \left(\frac{5}{2}\right)^3 = \frac{125}{8}$, arguing that "when inverting the fraction to remove the negative exponent, the minus sign of the base also disappears".

For each of the three cases:

1. a) Explain what conceptual mistake Sonia makes and which property she has applied incorrectly.
2. b) Show the correct step-by-step development and calculate the final numerical result as an integer or irreducible fraction.

Given the following numerical expressions:

1. a) Without previously calculating the exact numerical value of any of them, make a reasoned estimate answering these questions:

• Which of these powers will give a result strictly greater than $1$?

• Which will give a result between $0$ and $1$?

• Will any of them give a negative result? Justify your deduction based on the sign of the base, the parity of the exponent, and the meaning of the negative exponent as a multiplicative inverse.

1. b) Now calculate the exact value of each one in the form of an irreducible fraction or integer and check whether your initial estimates were correct.

Simplify the following expression, first factorising the composite bases and applying the laws of powers with integer exponents. Leave the result as a product of powers of prime numbers:

$\frac{12^3 \cdot 18^{-2}}{6^4 \cdot 2^{-5}}$

In a strategy video game, the number of soldiers in your faction doubles every hour of game play. If you start the game with $150$ soldiers, write a mathematical expression using powers indicating the number of soldiers after $h$ hours. How many soldiers will there be after $6$ hours?

In a chemical laboratory, a toxic substance contained in a liquid is being filtered. Each filter installed in the circuit reduces the amount of toxins to one third. If the initial amount of toxins is $81$ grams, express as a power with a negative exponent the amount of toxin remaining after passing through $5$ successive filters and calculate its exact value.

Calculate and fully simplify the following expression, giving the final result as an irreducible fraction:

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

In a mathematics exam, a student solved a calculation by writing $2^{-3} \cdot 5^{-3} = 10^{-6}$. Explain in detail the student's conceptual error, state the correct law that should have been used, and calculate the true result as a fraction.

A tennis tournament is played as a knockout competition (in each match, the loser is eliminated and the winner proceeds to the next round). If $2^7$ players participate in the first round, how many matches in total will have been played when the tournament finishes and there is a single overall winner? Express your reasoning using operations with powers.

Simplify the following expression containing fractions and negative exponents, applying the index laws step by step and giving the final result as an irreducible fraction without negative exponents:

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

Find the integer value of $x$ that satisfies the following exponential equation, expressing all terms as powers of the same base:

$5^{x+2} \cdot 25^{-1} = 125^3$

In a cleanroom at the Instituto de Microelectrónica de Barcelona (IMB-CNM, CSIC), the presence of microscopic impurities that can ruin microchips is monitored. A sensor detects a type $X$ environmental dust particle with a mass of $2^{-8}\text{ mg}$, and a type $Y$ ultrafine particle with a mass of $8^{-3}\text{ mg}$.

1. a) Express the mass of impurity $Y$ as a single base-$2$ power with an integer exponent.
2. b) Which of the two particles has a greater mass? Justify your answer by reasoning about the meaning of a negative integer exponent, without needing to calculate the decimal expression of either of them.
3. c) How many type $Y$ impurities are needed exactly to equal the mass of a single type $X$ particle? Set up the operation as a quotient of base-$2$ powers, apply the properties of powers with integer exponents, and express the result as an integer.

We want to design a mathematical equality with the following structure:

where $a$ and $b$ must be integers.

1. a) Factor the bases into prime factors ($2$ and $3$) and use the properties of powers with integer exponents to express the entire left-hand side as a single power of base $\frac{2}{3}$.
2. b) Write the fraction on the right-hand side, $\frac{32}{243}$, as a power of base $\frac{2}{3}$ with an integer exponent.
3. c) From the previous sections, obtain a linear equation that $a$ and $b$ must satisfy. Then, find two different pairs of integers $(a, b)$ that satisfy the equality, requiring that in at least one of them, one of the exponents is strictly negative.

Fully simplify the following algebraic expression, where $a$, $b$ and $c$ are non-zero real numbers. Apply the index laws and leave the final result expressed using positive exponents only:

$\frac{(a^{-2} \cdot b^3)^{-3} \cdot (a \cdot b^{-1} \cdot c^2)^4}{(a^3 \cdot c^{-5})^{-2}}$

A company invests a capital of $5000$ euros in an investment fund offering a compound interest rate of $4\%$ per annum. The correct mathematical formula to calculate the final capital after $t$ years is $C = 5000 \cdot (1,04)^t$. If due to a software error the system calculates the capital using a negative exponent, $C_{erroneo} = 5000 \cdot (1,04)^{-t}$, what financial meaning would this incorrect calculation have? Furthermore, deduce the exact mathematical relationship between the correct capital after $5$ years and the incorrect capital calculated for those same $5$ years.

Solve the following exponential equation. To do so, apply the laws of powers with integer exponents to split the bases and take out a common factor:

$3^{x+1} + 3^{x-1} - 3^x = 21$

A geometric fractal called the "Sierpinski Triangle" is constructed starting from a blue-filled equilateral triangle whose initial area is $1$ unit. In each iteration (step), each blue triangle is divided into four smaller equilateral triangles, and the central triangle is removed (leaving it empty/black). Look at the diagram illustrating the first few steps. Determine the mathematical expression, as a power, representing the remaining blue area after $n$ iterations. From that expression, what will be the exact blue area after $6$ iterations?

Prove mathematically that, for any real number $a \neq 0$ and any integer $n$, the identity $\left(\frac{1}{a}\right)^{-n} = a^n$ holds. In your proof, use the strict definition of a power with a negative exponent and the properties of operations with fractions.

Simplify the following algebraic expression, where $m$ and $n$ are arbitrary integers. After simplifying, check and justify whether the final result depends on the values taken by $m$ and $n$:

$\frac{4^{n+1} \cdot 10^{m-n}}{2^{m+n+2} \cdot 5^{m-n}}$

The intensity of a beam of light on passing through a certain type of tinted glass is reduced to $\frac{3}{4}$ of its incident value. If we stack $n$ of these identical glass panes, the resulting light intensity that manages to pass through is $I = I_0 \cdot \left(\frac{3}{4}\right)^n$, where $I_0$ is the initial intensity. Express the fraction of light that is lost (not the fraction that passes through) after passing through $3$ panes of glass, using powers with integer exponents for your calculations. In light of the result, determine whether the system blocks more or less than half of the original light.

Analyse and discuss with reasoning the resulting sign of the mathematical expression $(-a)^{-n}$ depending on two factors: whether the real number $a$ is strictly positive or negative, and whether the integer $n$ is even or odd. Draw up a chart or table showing the four possible cases resulting from this combination, accompanied by a specific numerical example for each case.

You can do it on paper and send a photo with your answer.

To simplify the following algebraic expression:

(with $x \ne 0$ and $y \ne 0$), two students propose two different working methods:

• Method 1 (Initial inversion): Rewrite each power with a negative exponent by moving the factors from the numerator to the denominator and vice versa ($a^{-n} = \frac{1}{a^n}$), to work only with positive exponents from the first step.

• Method 2 (Direct integer exponent arithmetic): Directly apply the power of a product and the power of a power by multiplying exponents with their sign, group powers with the same base by adding and subtracting integer exponents, and express with positive exponents only at the end if required.

1. a) Carry out the simplification step by step using Method 2, explicitly showing the operations with the signs of the exponents. Give the final result expressed using only positive exponents.
2. b) Justify why Method 1, although mathematically valid, is inefficient and generates a much higher risk of error in expressions with nested powers like this one.
3. c) Modify exclusively two of the numerical exponents appearing in the original expression $E$ so that the final simplified result is exactly $1$. Justify which ones you choose and what values you replace them with.

Consider the following algebraic function defined for any integer exponent $n \in \mathbb{Z}$:

1. a) Formally prove that the numerical value of $F(n)$ is constant (it does not depend at all on the value of $n$, whether it is positive, zero, or negative). To do so, factor out a suitable base-$3$ power in both the numerator and the denominator and simplify. Express the resulting constant as an irreducible fraction.
2. b) A student claims: "If we choose $n = -100$, the powers have very negative exponents and therefore are practically zero; thus, $F(-100)$ should tend to zero and not have the same value as for $n = 5$". Refute this argument with mathematical precision, distinguishing between the absolute magnitude of the terms and the relative quotient between them.
3. c) Inverse problem: Find what positive integer value the base $b$ ($b > 1$) must have so that the analogous expression:

takes exactly the numerical value $\frac{5}{12}$ for all $n \in \mathbb{Z}$.