The period of a pendulum, the growth of a laboratory culture every "half day", the relationship between the volume and the surface area of an animal: all these formulae use exponents that are not whole numbers. An exponent of $\frac{1}{2}$ or $\frac{2}{3}$ is really just a root written in a different way.
For the equivalences with rational exponents we will work with base $a>0$, an integer numerator and an integer index $n\ge2$. An even-index root is the principal root, which is non-negative. $0^q=0$ for $q>0$; $0^0$ and negative exponents of zero are not defined here. Odd roots allow negative radicands, for example $\sqrt[3]{-8}=-2$, but we do not simply carry all the rules over to negative bases: $\sqrt{(-1)^2}=1$, whereas $(-1)^{2/2}=-1$. In general, $\sqrt{x^2}=|x|$.
Practice★☆☆Not started
Express the following radicals as powers with fractional exponents: $\sqrt[5]{x^3}$, $\sqrt{7}$, $\sqrt[4]{a^5}$.
Express the following radicals as powers with fractional exponents: $\sqrt[5]{x^3}$, $\sqrt{7}$, $\sqrt[4]{a^5}$.
If the numerator and the denominator of the exponent have common factors, the radical can be simplified to a smaller index.
Practice★☆☆Not started
Simplify the following radicals by first expressing them as a power with a fractional exponent and simplifying the resulting fraction: $\sqrt[6]{8}$, $\sqrt[10]{x^5}$,…
Simplify the following radicals by first expressing them as a power with a fractional exponent and simplifying the resulting fraction: $\sqrt[6]{8}$, $\sqrt[10]{x^5}$, $\sqrt[12]{a^8}$.
With fractional exponents, multiplying or dividing radicals of different indices is as easy as adding or subtracting fractions.
Practice★☆☆Not started
Carry out the following multiplication, having first expressed the radicals as powers with fractional exponents. Give the final result in radical form:…
Carry out the following multiplication, having first expressed the radicals as powers with fractional exponents. Give the final result in radical form: $\sqrt{x} \cdot \sqrt[3]{x}$.
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:…
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: $\sqrt{2}$, $\sqrt[3]{3}$, $\sqrt[4]{4}$.
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…
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.
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…
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.
Back to the beginning: If a bacterial culture doubles every day ($2^t$), the growth factor after half a day ($t = 1/2$) is $2^{1/2} = \sqrt{2} \approx 1{,}41$, that is, it increases by approximately $41\,\%$.
Summary
Let's practice! 10 medium exercises in this section