Properties of Exponents

Express as a single power with base $5$ by applying the properties of powers: $\frac{5^4 \cdot 5^{-2}}{5^3}$.

Reduce to a single power using the appropriate properties: $\left((-7)^3\right)^{-4} \cdot (-7)^5$.

Calculate the value of the following power and express the result as an irreducible fraction: $\left(\frac{2}{3}\right)^{-3}$.

Express the number $\frac{1}{128}$ as a power with base $2$ using a negative exponent.

Write as a single power by applying the properties of operations with powers of the same exponent: $15^4 : 3^4 \cdot 2^4$.

State whether the following statement is true or false, justifying your answer using the properties of powers: "Double $2^8$ is $4^8$".

Simplify the following algebraic expression leaving the result with positive exponents: $\frac{x^3 \cdot y^{-2}}{x^{-1} \cdot y^4}$.

Simplify the following algebraic expression fully, using the properties of powers and leaving the result without negative exponents: $\left( \frac{a^{-2} \cdot b^3}{a \cdot b^{-1}} \right)^{-2}$.

Simplify the following algebraic expression, where $x$ and $y$ are non-zero real numbers and $a, b$ are positive integers:

$\left( \frac{x^{a-b} \cdot y^{b+a}}{x^{a+b} \cdot y^{b-a}} \right)^{-1}$

Apply the properties of powers with the same base and express the final result using only positive exponents.

Without using a calculator, determine which of the following quantities is greater: $3^{50}$ or $9^{20}$. Justify your answer by transforming one of the powers so that both have the same base.

Simplify the following expression by factorizing the bases first and applying the properties of powers. Express the result as a product of powers with prime bases: $\frac{18^2 \cdot 12^{-3}}{24^{-1} \cdot 9^3}$.

Simplify the following expression in which the exponents depend on the parameter $n$, taking out a common factor if necessary to be able to operate: $\frac{3^{n+2} - 3^n}{3^{n+1} + 3^n}$.

A student has shown the following algebraic working to simplify an expression:

$\frac{10^4 \cdot 5^{-2}}{2^4 \cdot 5^2} = \frac{(2 \cdot 5)^4 \cdot 5^{-2}}{2^4 \cdot 5^2} = \frac{2^4 \cdot 5^4 \cdot 5^{-2}}{2^4 \cdot 5^2} = 5^{4 - 2 - 2} = 5^0 = 1$

Identify if there is any error in their reasoning, explain it in detail and provide the correct result by strictly applying the properties of powers.

Find the value of the exponent $x$ so that the following equality holds: $2^x \cdot 2^3 = 2^{-1}$.

Solve the following exponential equation by expressing all terms as powers of the same base and operating with their exponents: $4^x \cdot 2^{x+1} = \frac{1}{8}$.

Find the values of $x$ and $y$ that simultaneously satisfy the following equalities, using the properties of powers to transform the system into a system of linear equations:

$2^x \cdot 4^y = 256$

$\frac{3^{2x}}{9^y} = 27$

A culture of bacteria in a biology laboratory doubles every hour. If initially there are $1000$ bacteria, express as a product with a power the number of bacteria there will be after $12$ hours and calculate its final value.

A radioactive substance reduces its mass to a third every day that passes. If today we have a sample of $81$ ounces, express using a power with base $3$ the amount of substance that will remain in $6$ days and calculate the result.

A capital is invested at an annual compound interest such that each year the money is multiplied by $1,05$. If the initial investment is $\$2000$, write the expression in the form of a power that calculates the capital after $t$ years. How much money will there be after $4$ years?

Moore's empirical law suggests that the number of transistors in a microprocessor doubles approximately every $2$ years. If in the year 1970 a processor had $2000$ transistors, express the number of transistors in the year $1970 + t$ using a power with base $2$. Use this property to estimate how many transistors a processor would have in the year 2010.

Imagine you have a sheet of paper $0,1\text{ mm}$ thick. Every time you fold the sheet in half on itself, its thickness doubles. Express the thickness of the sheet after $n$ folds using a power. If the average distance from the Earth to the Moon is approximately $384\,000\text{ km}$, set up an equation using powers of $2$ to estimate how many folds would be needed for the thickness of the paper to reach the Moon (you can estimate the value knowing that $2^{10} \approx 10^3$).

A packaging company manufactures cubic boxes. If they decide to triple the edge length of a standard box, by what factor will its volume be multiplied? Express your reasoning using the properties of the power of a product.

In astrophysics, the intensity of the light we receive from a star is inversely proportional to the square of the distance we are from it. If we move to a distance $4$ times greater from the star, express using a power with a negative exponent the factor by which the original light intensity is multiplied. If the initial intensity was $I_0$, what will the new intensity be in terms of $I_0$?

The Sierpinski carpet is a fractal constructed from a square. In each iteration, each solid square is divided into $9$ equal squares and the central one is removed. If the initial area ($n=0$) is $1\text{ m}^2$, in the first iteration the remaining area is $\frac{8}{9}\text{ m}^2$. Express the remaining area after $n$ iterations using powers. Calculate the area that will remain after $5$ iterations.