Exercises: Roots (powers with rational exponents)

Express the following radicals as powers with fractional exponents: $\sqrt[5]{x^3}$, $\sqrt{7}$, $\sqrt[4]{a^5}$.

Write the following powers with fractional exponents in radical form: $m^{2/3}$, $5^{1/4}$, $y^{-3/2}$.

Calculate the exact numerical value of the following powers with rational exponents without using a calculator: $27^{1/3}$, $16^{3/4}$, $25^{-1/2}$.

Simplify the following radicals by first expressing them as a power with a fractional exponent and simplifying the resulting fraction: $\sqrt[6]{8}$, $\sqrt[10]{x^5}$, $\sqrt[12]{a^8}$.

Carry out the following multiplication, having first expressed the radicals as powers with fractional exponents. Give the final result in radical form: $\sqrt{x} \cdot \sqrt[3]{x}$.

Carry out the following division by expressing the terms in the form of a power with a rational exponent and simplify the result: $\frac{\sqrt[4]{a^3}}{\sqrt{a}}$.

Calculate the power of the root by writing the expression with a fractional exponent and applying the laws of indices: $(\sqrt[5]{2})^3$.

Express the following operation as a single power with a rational exponent: $x^{1/2} \cdot x^{1/3} : x^{1/4}$.

In a 4th-year ESO exam, Marcos has to express several radicals as powers. When he reaches the expression $\sqrt[3]{7^2}$, he writes on his paper:

When the teacher asks him why he solved it that way, Marcos argues: "3 is the first number you read when saying cube root, so it must be placed in the numerator of the exponent's fraction".

1. a) Explain in a reasoned manner what Marcos's conceptual error is, recalling the relationship between the index of a root and the fractional exponent when raising the expression to that index.
2. b) Correct Marcos's equality by writing the power with a fractional exponent that truly corresponds to $\sqrt[3]{7^2}$.
3. c) Apply the correct rule to calculate the exact numerical value of $\sqrt[3]{64^2}$, first finding the root of the radicand and then raising the result.

In the Doñana National Park, the veterinary team monitoring the Iberian lynx uses Kleiber's allometric law to estimate the basal metabolic rate $B$ (the minimum daily energy at rest needed by the animal to live, measured in kilocalories per day) as a function of its body mass $M$ (in kilograms):

1. a) Express the formula for $B$ by replacing the power with a fractional exponent with its equivalent radical form.
2. b) To calculate the basal energy of a young lynx of $M = 16\text{ kg}$, a volunteer proposes calculating first $16^3 = 4096$ and then trying to extract the fourth root $\sqrt[4]{4096}$, whereas a veterinary technician proposes calculating first $\sqrt[4]{16}$ and then cubing that value. Calculate the final value of $B$ following the technician's procedure and justify why her method is much more convenient for doing the calculation by hand without a calculator.

In a laboratory culture, the mass of a fungus is modelled by the expression $M(t) = M_0 \cdot 2^{t/3}$, where $t$ is the time elapsed in days. If the initial mass is $15$ grams, calculate the mass of the fungus after $6$ days and express it in radical form for the time $t = 4$ days.

Evaluate and simplify the following expression as far as possible, using the laws of indices for rational exponents: $\sqrt[3]{a^2 \cdot \sqrt{a}}$.

The period $T$ (in seconds) of a simple pendulum of length $L$ (in metres) is given by the formula $T = 2\pi \cdot \left(\frac{L}{g}\right)^{1/2}$, where $g$ is the acceleration due to gravity. If we want to multiply the period of oscillation by $3$, by what factor must the length of the pendulum be multiplied? Reason your answer using the laws of indices.

Simplify the following algebraic expression working with fractional exponents and give the final result without negative or fractional exponents: $\left( \frac{x^{-1/2} \cdot y^{2/3}}{x^{1/3} \cdot y^{-1/6}} \right)^6$.

A cubic tank contains exactly $120$ litres of water when completely full. Express the length of its internal edge in decimetres using a power with a fractional exponent.

Rewrite with a common index and order the following numbers from smallest to largest, having first converted each radical to a power with a rational exponent: $\sqrt{2}$, $\sqrt[3]{3}$, $\sqrt[4]{4}$.

Carry out the following calculation, having first converted all radicals into powers with a rational exponent and base $3$. Simplify the result and express it in radical form: $\frac{\sqrt[4]{27} \cdot \sqrt{3}}{\sqrt[3]{9}}$.

In economics, if a principal grows at an annual compound interest rate of $r$, the final amount after $t$ years is calculated using the formula $C = C_0 (1+r)^t$. Calculate the final amount if $2\,000$ € is invested at an annual interest rate of $5$ % for $18$ months (express the time as a fraction of a year and leave the calculation expressed with a rational exponent).

Two 4th-year ESO students, Lucía and David, need to simplify and find the exact value of the following operation involving radicals:

• Lucía decides to work directly with radicals: she finds the least common multiple of the indices to reduce them to a common index ($6$) and operates within a single root.

• David decides to convert everything to powers: he factors the radicand into prime factors ($16 = 2^4$), expresses each root as a power with base $2$ and a fractional exponent, and adds the exponents.

1. a) Solve the operation step by step following Lucía's radical method until you obtain the final numerical value.
2. b) Solve the operation step by step following David's fractional exponent method until you obtain the same numerical value.
3. c) Compare both approaches: which of the two methods do you consider more direct or less prone to errors when the radicands are powers of the same base? Justify your answer.

In the development programs at the High Performance Centre (CAR) in Madrid, a biomechanical scaling model is applied to estimate the relative strength index $F$ of a junior weightlifter as a function of their body mass $M$ (in kilograms):

During their training phase, the athlete goes from an initial mass of $27\text{ kg}$ to reaching $64\text{ kg}$ after several seasons of training.

1. a) Estimate and reason before calculating: the athlete's body mass has more than doubled ($64/27 \approx 2,37$). Taking into account that the fractional exponent is less than $1$ ($2/3 < 1$), do you expect their relative strength $F$ to double, more than double, or less than double? Justify your reasoning.
2. b) Transform the formula into radical form and calculate the exact value of $F$ for $M = 27\text{ kg}$ and for $M = 64\text{ kg}$, showing each step in detail.
3. c) Check whether your estimation in section a) was correct by calculating the ratio between both strengths ($F_{64}/F_{27}$) and briefly explain why strength sports are divided into such strict weight categories.

Kepler's third law states that the square of the orbital period $T$ of a planet is proportional to the cube of the mean radius of its orbit $R$. Express $T$ in terms of $R$ and the constant of proportionality $k$, using fractional exponents. If the radius of Jupiter's orbit is approximately $5,2$ times the radius of Earth's orbit, how many Earth years does Jupiter take to complete one full orbit around the Sun?

Simplify the following algebraic expression by working from the inside out and using powers with rational exponents. Express the answer as a single power of base $x$: $\sqrt{x \cdot \sqrt[3]{x^2 \cdot \sqrt[4]{x^{-1}}}}$.

The body surface area $S$ and the volume $V$ of a mammal are approximately related by the allometric law $S = k \cdot V^{2/3}$, where $k$ is a constant. If an elephant has a volume $1\,000$ times greater than that of a medium-sized dog, how many times greater will the elephant's body surface area be compared to the dog's? Justify your answer step by step using the properties of powers.

Solve the following exponential equation. To do so, first express both sides of the equation as powers with base $5$ and rational exponents: $\sqrt[3]{5^{2x-1}} = \frac{1}{\sqrt{5^{x+2}}}$.

Rationalise the expression $\frac{2a}{\sqrt[5]{a^2}}$ by interpreting the fraction as a quotient of powers with rational exponents. Show all the steps using index laws and express the final result in radical form.

Prove that the property of radicals $\sqrt[n]{\sqrt[m]{a}} = \sqrt[n \cdot m]{a}$ holds true using exclusively the definition of a root as a power with a rational exponent and the properties of powers.

In the theory of special relativity, the Lorentz factor is defined by the expression $\gamma = \left(1 - \frac{v^2}{c^2}\right)^{-1/2}$, where $v$ is the speed of the spacecraft and $c$ is the speed of light. Express this formula using radical and fraction notation. Next, calculate the exact value of $\gamma$ as a simplified fraction if a spacecraft travels at a speed of $v = \frac{\sqrt{3}}{2}c$.

Simplify the following algebraic expression by factorising out the common factor with the lowest exponent (using powers with rational exponents): $x^{5/2} - x^{3/2}$. Next, evaluate the numerical value of the simplified expression for $x = 4$.

An educational programming team is designing a mathematical simulator. The system must transform any positive real number $x$ into the exact result $x^{5/6}$. However, the graphics processor only allows chaining two consecutive roots with internal powers, following the structure:

where the parameters $a, b, c, d$ must be positive integers, with valid root indices ($a \ge 2$ and $c \ge 2$).

1. a) Express the entire structure as a single power with base $x$ and deduce the equation that the parameters $a, b, c, d$ must satisfy for the result to match $x^{5/6}$ exactly.
2. b) Find two completely different combinations of integer numerical values for $(a, b, c, d)$ that satisfy all the conditions imposed by the processor.
3. c) A novice programmer claims that if the two root indices ($a$ and $c$) and the two exponents ($b$ and $d$) are swapped, the final result of the algorithm will change. Formally prove, using the properties of powers with rational exponents and the properties of multiplication, whether the programmer is right or wrong.

The Spanish Institute of Oceanography (IEO) monitors mussel growth on the raft platforms of the Ría de Arousa (Galicia). To estimate the commercial meat mass $M$ (in grams) from the total shell length $L$ (in centimetres), biologists use the allometric model:

1. a) Convert the decimal exponent $2,5$ into an irreducible fraction and write the complete formula using a single simplified radical.
2. b) A farmer claims that if an adult mussel is four times the length of a juvenile ($L_{\text{adult}} = 4 \cdot L_{\text{juvenile}}$), its mass will also be four times greater. Prove algebraically, applying the properties of powers with rational exponents, by what exact factor the mass is multiplied when the shell length is quadrupled.
3. c) Calculate the exact mass predicted by the model for a small specimen with $L = 4\text{ cm}$ and for a large one with $L = 9\text{ cm}$, showing the intermediate steps with roots and integer powers before operating with the factor $0,25$. Based on the results and the relationship between linear measurements (1D) and volumes (3D), justify in a reasoned way why in living beings mass increases at a much faster rate than length.

Unit Challenge. How much energy powers our country and what part corresponds to your home? Become an energy analyst and prepare a brief impact report.

1. Official data: Go to the Red Eléctrica de España (REE) website and check the recent annual peninsular electricity demand in MWh. Convert it to joules knowing that $1\text{ kWh} = 3,6 \cdot 10^6\text{ J}$ and express it in scientific notation with two decimal places.

2. From macro to micro: Take the monthly consumption from a real electricity bill of your home (or the Spanish average of $270\text{ kWh/month}$). Using operations with base-$10$ powers, calculate how many homes like yours the national demand supplies.

3. Calculation with rational exponent: Also obtain the demand from $5$ years ago ($V_0$) and the current one ($V_f$). In a spreadsheet, determine the annual growth rate using the fractional exponent formula $r = \left(\frac{V_f}{V_0}\right)^{\frac{1}{5}} - 1$.

4. Final proposal: Write a brief report comparing magnitudes and proposing a viable energy-saving measure at home.

Assessment criteria:

• Correctness in unit conversions and in scientific notation.

• Precision in operations with powers and in calculations with roots/fractional exponents.

• Rigorous extraction of real data from the official website and clarity of the report.