Properties of Powers

Express as a single power of base $5$ by applying index laws: $\frac{5^4 \times 5^{-2}}{5^3}$.

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

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 of base $2$ using a negative exponent.

In astrophysics, the intensity of light received from a star is inversely proportional to the square of the distance 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 $I_0$?

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

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

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

Fully simplify the following algebraic expression, using index laws and leaving the result without negative exponents: $\left( \frac{a^{-2} \times b^3}{a \times b^{-1}} \right)^{-2}$.

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

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

Apply the index laws for powers with the same base and express the final result using positive exponents only.

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

A student carried out the following algebraic working to simplify an expression:

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

Identify whether there is an error in their reasoning, explain it in detail, and provide the correct result by strictly applying index laws.

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

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

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 power of a product law.

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

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

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

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

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

A rumour in a school spreads by doubling every hour the number of people who know it. If $1000$ people initially know it, express as a product with a power the number of people who will know it after $12$ hours, and calculate its final value.

A radioactive substance reduces its mass to one-third every passing day. If today we have a sample of $81$ grams, express using a power of base $3$ the amount of substance remaining in $6$ days' time, and calculate the result.

A sum of money is invested at an annual compound interest rate such that each year the money is multiplied by $1.05$. If the initial investment is $40\,000$ pesos, write the expression in index form that calculates the capital after $t$ years. How much money will there be after $4$ years?

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 $1\text{ m}^2$, after 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 remaining after $5$ iterations.

Moore's empirical law suggests that the number of transistors in a microprocessor doubles approximately every $2$ years. If in 1970 a processor had $2000$ transistors, express the number of transistors in $1970 + t$ using a power of 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 with a thickness of $0.1\text{ mm}$. Every time you fold the sheet in half over itself, its thickness doubles. Express the thickness of the sheet after $n$ folds as a power. If the mean 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 may estimate the value knowing that $2^{10} \approx 10^3$).