Exercises: Laws of indices

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

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

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

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

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

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

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

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

Mateo has to simplify the expression $2^4 \cdot 5^4$. On his exam he writes the following procedure:

and justifies his answer saying: "I multiply the bases ($2 \cdot 5 = 10$) and add the exponents ($4 + 4 = 8$)".

1. a) Explain in detail why Mateo's reasoning violates the meaning of a power as repeated multiplication of factors and which rule he mixed up by mistake.
2. b) Correct the exercise by applying the appropriate property and calculate the final numerical value.

Before doing any calculations with pencil and paper, estimate which of these two numbers you think is larger: $A = 2^{30}$ or $B = 3^{20}$. Briefly justify your initial intuition, taking into account that $A$ has a greater exponent but a smaller base than $B$.

Next, verify your prediction rigorously without a calculator: use the power of a power property, $(a^m)^n = a^{m \cdot n}$, to rewrite both numbers so that they have the same outer exponent and conclude which one is larger.

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

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.

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

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.

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.

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

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 $2000$ €, write the expression in index form that calculates the capital after $t$ years. How much money will there be after $4$ years?

To simplify the expression $\frac{6^6}{2^4 \cdot 3^4}$, two students propose different paths:

• Elena first groups the factors in the denominator by applying the property of powers with the same exponent.

• Pablo decomposes the base of the numerator into prime factors and applies the power of a product.

1. a) Develop the step-by-step simplification following Elena's method and, subsequently, following Pablo's.
2. b) Compare both procedures: which one is faster and has less risk of error in this case? Would your choice of method change if the initial expression had been $\frac{6^6}{2^4 \cdot 3^5}$? Justify your answer.

The Instituto Nacional de Ciberseguridad (INCIBE), based in León, analyzes a protocol for broadcasting urgent alerts in educational centers. In an initial phase ($n = 0$), the alert is sent to the school management. In each successive phase $n$, the number of newly informed recipients grows exponentially following the expression $2^n$.

1. a) If in phase $n+3$ the number of new recipients exceeds that of phase $n$ by $112$ people, write down the corresponding equation.
2. b) Solve this equation by factoring out the lowest power of base $2$ to determine the value of phase $n$.
3. c) Applying only the quotient of powers with the same base, calculate how many times greater the volume of new recipients is in phase $n+3$ compared to phase $n$.

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: $\frac{3^{n+2} - 3^n}{3^{n+1} + 3^n}$.

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.

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

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:

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

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

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 $I_0$?

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

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 $2^{10} \approx 10^3$).

Design a fractional expression of the form $\frac{a^x \cdot b^y}{c^z}$ that simultaneously satisfies these four conditions:

1. The bases $a$, $b$, and $c$ are distinct composite numbers belonging to the set $\{4, 6, 8, 9, 12\}$.

2. The exponents $x$, $y$, and $z$ are non-zero integers, and at least one of them must be negative.

3. No final exponent of the powers can be $0$.

4. After decomposing the bases into prime factors and applying the properties of powers, the resulting irreducible fraction must be exactly $\frac{3}{2}$.

Show the entire deductive process you followed to find the bases and the exponents using the equations of the exponents of the prime bases $2$ and $3$.

During a review session, David states: "Since $2 \cdot 5 = 10$, any number generated by the expression $2^{n+2} \cdot 5^n$, for any positive integer $n$, must necessarily end in two zeros, because the exponent of $2$ has a $+2$".

1. a) Estimate and calculate the value of the expression for $n = 1$, $n = 2$, and $n = 3$. Is David's general statement correct?
2. b) Algebraically transform the expression $2^{n+2} \cdot 5^n$ using the properties of powers to write it in the form $k \cdot 10^n$, where $k$ is an integer independent of $n$.
3. c) Relying on your work in section b), explain what David's conceptual error is, how many zeros the number actually has as a function of $n$, and what its first significant digit always is.