An expression of the form $\$\sqrt[n]{a}\$, where \$n\$ is the index and \$a\$ is the radicand.
NoteThe process of simplifyinga square root is the reverseof expandinga square root.
Simplifying Higher-Order Radicals
The process for simplifying higher-order radicals $\$\sqrt[n]{a}\$ is similar to that for square roots:
- Factor the radicand \$a\$ into its prime factors.
- Group the factors into groups of size \$n\$.
- Move each group of size \$n\$ outside the radical.
The process of simplifyinga higher-order radical is the reverseof expandinga higher-order radical.
Operations with Radicals
Multiplying Radicals
NoteThe indicesof the radicals must be the sameto multiply them.
To multiply two radicals $\$\sqrt[n]{a}\$ and $\$\sqrt[n]{b}\$ with the same index \$n\$:
- Multiply the radicands: \$a \cdot b\$.
- Keep the same index: \$n\$.
The product of two radicals is simplifiedif the productof the radicands is simplified.
Dividing Radicals
NoteThe indicesof the radicals must be the sameto divide them.
To divide two radicals $\$\sqrt[n]{a}\$ and $\$\sqrt[n]{b}\$ with the same index \$n\$:
- Divide the radicands: \$\frac{a}{b}\$.
- Keep the same index: \$n\$.
The quotient of two radicals is simplifiedif the quotientof the radicands is simplified.
Adding and Subtracting Radicals
NoteThe indicesof the radicals and the radicandsmust be the sameto add or subtract them.
To add or subtract two radicals $\$\sqrt[n]{a}\$ and $\$\sqrt[n]{a}\$ with the same index \$n\$ and the same radicand \$a\$:
- Add or subtract the coefficients of the radicals.
- Keep the same index and radicand.
The sum or difference of two radicals is simplifiedif the coefficientsare simplified.
Rationalising the Denominator
The process of eliminating radicals from the denominator of a fraction.
To rationalise the denominator of a fraction \$\frac{a}{\sqrt{b}}\$:
- Multiply the numerator and the denominator by the conjugate of the denominator.
- Simplify the fraction.
The process of rationalisingthe denominator is the reverseof simplifyinga fraction with a radical in the denominator.
Self review1. Simplify the radical $\$\sqrt{50}\$. 2. Multiply the radicals $\$\sqrt{2}\$ and $\$\sqrt{8}\$. 3. Divide the radicals $\$\sqrt{18}\$ and $\$\sqrt{2}\$. 4. Add the radicals $\$\sqrt{3}\$ and \$2\sqrt{3}\$. 5. Rationalise the denominator of the fraction \$\frac{5}{\sqrt{3}}\$.
