Example 1
Completion requirements
| Example 1 |
Simplify \(\sqrt{20}\).
Method 1: Prime factorization
Step 1: Rewrite the radicand as a product of prime factors.
\(\begin{align}
\sqrt{20} &= \sqrt{2 \cdot 2 \cdot 5} \\
&= \sqrt{2^2 \cdot 5}\\
\end{align} \)
Step 2: Rewrite the radical as a product of the perfect square radical and the non-perfect square radical.
\( \sqrt{2^2 \cdot 5} = \sqrt{2^2} \cdot \sqrt{5}\)
Step 3: Simplify.
\(\begin{align}
\sqrt{2^2 \cdot 5} &= 2 \cdot \sqrt{5} \\
&= 2\sqrt{5} \\
\end{align}\)
Therefore, \(\sqrt{20} = 2\sqrt{5}\).
Method 2: Perfect squares (because the index is 2)
Step 1: Rewrite the radicand as a product of factors, looking specifically for factors that are perfect squares.
\( \sqrt{20} = \sqrt{4 \cdot 5}\)
Step 2: Rewrite the radical as a product of the perfect square radical and the non-perfect square radical.
\(\begin{align}
\sqrt{4 \cdot 5} &= \sqrt{4} \cdot \sqrt{5} \\
&= \sqrt{2^2} \cdot \sqrt{5}\\
\end{align}\)
Step 3: Simplify.
\(\begin{align}
\sqrt{2^2 \cdot 5} &= 2 \cdot \sqrt{5} \\
&= 2\sqrt{5} \\
\end{align}\)
Therefore, \(\sqrt{20} = 2\sqrt{5}\).
Note that not all entire radicals can be simplified and written as mixed radicals. The radicand must contain perfect \(n^{th}\) root factors in order for it to be rewritten as a mixed radical. For example, given \(\sqrt{55}\) , the factors of \(55\) are \(1 \times 55\) and \(5 \times 11\). Notice that neither pair of factors contains a perfect square. Therefore, \(\sqrt{55}\) is already written in its simplest form.
Step 1: Rewrite the radicand as a product of prime factors.
\(\begin{align}
\sqrt{20} &= \sqrt{2 \cdot 2 \cdot 5} \\
&= \sqrt{2^2 \cdot 5}\\
\end{align} \)
Step 2: Rewrite the radical as a product of the perfect square radical and the non-perfect square radical.
\( \sqrt{2^2 \cdot 5} = \sqrt{2^2} \cdot \sqrt{5}\)
Step 3: Simplify.
\(\begin{align}
\sqrt{2^2 \cdot 5} &= 2 \cdot \sqrt{5} \\
&= 2\sqrt{5} \\
\end{align}\)
Therefore, \(\sqrt{20} = 2\sqrt{5}\).
Method 2: Perfect squares (because the index is 2)
Step 1: Rewrite the radicand as a product of factors, looking specifically for factors that are perfect squares.
\( \sqrt{20} = \sqrt{4 \cdot 5}\)
Step 2: Rewrite the radical as a product of the perfect square radical and the non-perfect square radical.
\(\begin{align}
\sqrt{4 \cdot 5} &= \sqrt{4} \cdot \sqrt{5} \\
&= \sqrt{2^2} \cdot \sqrt{5}\\
\end{align}\)
Step 3: Simplify.
\(\begin{align}
\sqrt{2^2 \cdot 5} &= 2 \cdot \sqrt{5} \\
&= 2\sqrt{5} \\
\end{align}\)
Therefore, \(\sqrt{20} = 2\sqrt{5}\).
Note that not all entire radicals can be simplified and written as mixed radicals. The radicand must contain perfect \(n^{th}\) root factors in order for it to be rewritten as a mixed radical. For example, given \(\sqrt{55}\) , the factors of \(55\) are \(1 \times 55\) and \(5 \times 11\). Notice that neither pair of factors contains a perfect square. Therefore, \(\sqrt{55}\) is already written in its simplest form.