Operations with irrational numbers
In physics and engineering, formulas often leave radicals half-simplified: the period of a pendulum, the tension in a cable, the volume of a crystal. Knowing how to work with $\sqrt{\phantom{x}}$ without reaching for the calculator at every step is what lets you arrive at an exact, simplified expression.
Taking factors out of a radical
Break the radicand down into prime factors and take out of the radical those that can be grouped in sets whose size equals the index.
Take out all possible factors from the following radicals to express them in their simplest form: a) $\sqrt{252}$ b) $\sqrt[3]{1080}$
Take out all possible factors from the following radicals to express them in their simplest form:
- a) $\sqrt{252}$
- b) $\sqrt[3]{1080}$
Adding and subtracting like radicals
You can only reduce to a single term by adding coefficients radicals with the same index and the same radicand (just as you can only add like terms in algebra).
Carry out the following calculation by collecting like radicals and express the final result: $7\sqrt{5} - 2\sqrt{5} + 4\sqrt{5} - \sqrt{5}$
Carry out the following calculation by collecting like radicals and express the final result:
$7\sqrt{5} - 2\sqrt{5} + 4\sqrt{5} - \sqrt{5}$
Multiplying, dividing and raising radicals to a power
With an integer index $n\ge2$, use the formulas below with radicands $a,b\ge0$ if $n$ is even; for an odd index they can be real numbers of any sign. In quotients you need $b\ne0$. For the power, $m$ is a positive integer; if it is negative, you also need $a\ne0$.
For positive radicands, two different indices are first reduced to a common index (the lowest common multiple of the indices):
Multiply the following radicals and simplify the result by taking out any possible factors: $\sqrt{15} \cdot \sqrt{10}$
Multiply the following radicals and simplify the result by taking out any possible factors:
$\sqrt{15} \cdot \sqrt{10}$
Express as a single radical using the laws of roots and simplify the result: $\sqrt{\sqrt[3]{64}}$
Express as a single radical using the laws of roots and simplify the result:
$\sqrt{\sqrt[3]{64}}$
Carry out the following combined operation with radicals by applying the distributive property, and simplify the result as much as possible:…
Carry out the following combined operation with radicals by applying the distributive property, and simplify the result as much as possible:
$\sqrt{3} \cdot (\sqrt{27} - \sqrt{12}) + 5\sqrt{9}$
Rationalising: removing the root from the denominator
Dividing by a number containing roots is awkward. Rationalising means transforming the fraction so that the denominator has no radicals, by multiplying the numerator and denominator by a suitable expression.
Rationalise the denominator of the following fraction and simplify the resulting expression: $\frac{18}{\sqrt{3}}$
Rationalise the denominator of the following fraction and simplify the resulting expression:
$\frac{18}{\sqrt{3}}$
The intensity of an electromagnetic signal at a certain point in space has been modelled using the expression $I = \frac{12}{\sqrt{5} - \sqrt{2}}$ W/m$^2$. Rationalise…
The intensity of an electromagnetic signal at a certain point in space has been modelled using the expression $I = \frac{12}{\sqrt{5} - \sqrt{2}}$ W/m$^2$. Rationalise the expression to calculate the exact value of the intensity without roots in the denominator.
Show that the following expression represents an integer, carrying out the fraction additions and necessary rationalisations step by step:…
Show that the following expression represents an integer, carrying out the fraction additions and necessary rationalisations step by step:
$\frac{\sqrt{7} + \sqrt{3}}{\sqrt{7} - \sqrt{3}} + \frac{\sqrt{7} - \sqrt{3}}{\sqrt{7} + \sqrt{3}}$