Integer Exponents

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

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

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.

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

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

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

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

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: $4^{-3} \times 2^{-3} \times 5^{-3}$.

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 $3$: $9^2 \times 27^{-1} \times 3^4$.

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:

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

In a mathematics exam, a student solved a calculation by writing $2^{-3} \times 5^{-3} = 10^{-6}$. 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.

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:

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

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:

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

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 $n$:

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

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.

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.

A company invests a capital of $100\,000$ pesos in an investment fund offering a compound interest rate of $4\%$ per year. The correct mathematical formula to calculate the final capital after $t$ years is $C = 100\,000 \times (1,04)^t$. If due to a software error the system calculates the capital using a negative exponent, $C_{erroneo} = 100\,000 \times (1,04)^{-t}$, 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.

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?

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 $I = I_0 \times \left(\frac{3}{4}\right)^n$, 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.

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

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

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:

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