Radicals

Radicals are the opposite operations to exponent like how we can undo a multiplication by division, we can undo an exponent through a radical and vice versa.

Square Roots

Some important terms include...

1. The principal square root of $a$ is the nonnegative number that, when multiplied by itself, equals $a$.
2. A radical is the symbol ($\sqrt{}$) which is used to denote a root.
3. The radicand is the expression written inside a root like $a$ in $\sqrt{a}$.
4. A radical expression is an expression which includes a radical term.

_________ for Simplifying Square Roots	Definition
Product Rule	$\sqrt{ab} = \sqrt{a} \times \sqrt{b}$
Quotient Rule	$\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}$

note

We can simplify roots like $\sqrt{300}$ through these rules. Through the product rule this expression can be rewritten as $\sqrt{100 \times 3} = \sqrt{100} \times \sqrt{3}$. This gives us the simpliest form being $10\sqrt{3}$.

We can only add or subtract terms with the same radical. For example, the expression $5\sqrt{5} + 3\sqrt{5}$ can be simplified to $8\sqrt{5}$. Also, expressions like $\sqrt{300} + 5\sqrt{3}$ can be simplified too only when we simplify $\sqrt{300}$ to include the same radical. This gives us the expression $10\sqrt{3} + 5\sqrt{3}$ which simplifies to $15\sqrt{3}$.

Rationalizing Denominators

If a radical expression is written in simplest form then it cannot contain a radical in the denominator. The process of removing these radicals from the denominators of fractions is called rationalizing the denominator.

For single term denominators in the form $b\sqrt{c}$, multiply the fraction by $\frac{\sqrt{c}}{\sqrt{c}}$. It must be noted that $b$ can be equal to $1$.

For denominators that contain a sum of a rational and an irrational term in the form $a + b\sqrt{c}$, then we use the conjugate which is $a - b\sqrt{c}$. This means we multiply the fraction with $\frac{a - b\sqrt{c}}{a - b\sqrt{c}}$.

Finally, for denominators that contain a difference of a rational and an irrational term in the form $a - b\sqrt{c}$, then we use the conjugate which is $a + b\sqrt{c}$. This means we multiply the fraction with $\frac{a + b\sqrt{c}}{a + b\sqrt{c}}$.

note

We are multiplying these fractions by fractions that are equivalent to $1$ when we are simplifying. The identity property of multiplication shows us we can do this without changing the value of the original fraction. Also note, we obtained the conjugates by changing the sign of the radical portion of the denominator.

Rational Roots

1. If $a$ is a real number with at least one $n$th root, then the principal $n$th root of $a$ is the number with the same sign as $a$ that, when raised to the $n$th power, equals $a$. This root is written as $\sqrt[n]{a}$.
2. The index of the radical in a radical expression is the value of $n$ in $\sqrt[n]{a}$.

note

The expression $\sqrt{a}$ is equivalent to writing $\sqrt[2]{a}$.

Rational exponents are another way to express principal $n$th roots. The general form is $a^{\frac{m}{n}}$ which is equivalent to $(\sqrt[n]{a})^m$ and $\sqrt[n]{a^m}$. All three expressions are equivalent to each other.