Example 3
Completion requirements
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Example 3
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Simplify \(\sqrt[4]{{2\thinspace 592}}\).
Method 1: Prime factorization
Step 1: Rewrite the radicand as a product of prime factors.
\(\sqrt[4]{2\thinspace 592} = \sqrt[4]{{2 \cdot 2\cdot 2\cdot 2\cdot 2 \cdot 3\cdot 3\cdot 3\cdot 3}}\)
Step 2: Group as many identical factors as match the value of the index. Because the index is \(4\), group four factors of two and four factors of three, \(2 \cdot 2 \cdot 2\cdot 2\) and \(3\cdot 3\cdot 3\cdot 3\), and rewrite them as \(2^4\) and \(3^4\), perfect fourths.
\( \sqrt[4]{{{\color{red}2 \cdot 2 \cdot 2\cdot 2}\cdot 2\cdot {\color{red}3\cdot 3\cdot 3\cdot 3}}} = \sqrt[4]{{2^4 \cdot 2 \cdot 3^4}}\)
Step 3: Rewrite the radical as a product of the perfect fourth radicals and the non-perfect fourth radicals.
\( \sqrt[4]{{2^4 \cdot 2 \cdot 3^4}} = \sqrt[4]{{2^4}} \cdot \sqrt[4]{2} \cdot \sqrt[4]{3^4}\)
Step 4: Simplify.
\(\begin{align}
\sqrt[4]{{2^4}} \cdot \sqrt[4]{2} \cdot \sqrt[4]{3^4} &= 2 \cdot \sqrt[4]{2} \cdot 3 \\
&=6\sqrt[4]{2} \\
\end{align}\)
Therefore, \(\sqrt[4]{{2\thinspace 592}} = 6\sqrt[{4}]{2}\).
Method 2: Perfect fourth factors (because the index is 4)
Step 1: Rewrite the radicand as a product of factors, looking specifically for factors that are perfect fourths.
\(\begin{align}
\sqrt[4]{{2\thinspace 592}} &= \sqrt[4]{{16\cdot 81 \cdot 2}} \\
&= \sqrt[4]{{2^4 \cdot 3^4 \cdot 2}} \\
\end{align}\)
Step 2: Rewrite the radical as a product of the perfect fourth radicals and the non-perfect fourth radicals.
\( \sqrt[4]{{2^4 \cdot 3^4 \cdot 2}} = \sqrt[4]{{2^4}}\cdot \sqrt[4]{{3^4}} \cdot \sqrt[4]{2}\)
Step 3: Simplify.
\( \sqrt[4]{{2^4}}\cdot \sqrt[4]{{3^4}} \cdot \sqrt[4]{2} = 2\cdot 3\sqrt[4]{2}\)
Therefore, \(\sqrt[4]{{2\thinspace 592}} = 6\sqrt[4]{2}\).
From this point on, because of its efficiency, the perfect \(n^{th}\) root factors method will be the only one used in solutions. However, you are certainly free to choose a solution method that works best for you.
Step 1: Rewrite the radicand as a product of prime factors.
\(\sqrt[4]{2\thinspace 592} = \sqrt[4]{{2 \cdot 2\cdot 2\cdot 2\cdot 2 \cdot 3\cdot 3\cdot 3\cdot 3}}\)
Step 2: Group as many identical factors as match the value of the index. Because the index is \(4\), group four factors of two and four factors of three, \(2 \cdot 2 \cdot 2\cdot 2\) and \(3\cdot 3\cdot 3\cdot 3\), and rewrite them as \(2^4\) and \(3^4\), perfect fourths.
\( \sqrt[4]{{{\color{red}2 \cdot 2 \cdot 2\cdot 2}\cdot 2\cdot {\color{red}3\cdot 3\cdot 3\cdot 3}}} = \sqrt[4]{{2^4 \cdot 2 \cdot 3^4}}\)
Step 3: Rewrite the radical as a product of the perfect fourth radicals and the non-perfect fourth radicals.
\( \sqrt[4]{{2^4 \cdot 2 \cdot 3^4}} = \sqrt[4]{{2^4}} \cdot \sqrt[4]{2} \cdot \sqrt[4]{3^4}\)
Step 4: Simplify.
\(\begin{align}
\sqrt[4]{{2^4}} \cdot \sqrt[4]{2} \cdot \sqrt[4]{3^4} &= 2 \cdot \sqrt[4]{2} \cdot 3 \\
&=6\sqrt[4]{2} \\
\end{align}\)
Therefore, \(\sqrt[4]{{2\thinspace 592}} = 6\sqrt[{4}]{2}\).
Method 2: Perfect fourth factors (because the index is 4)
Step 1: Rewrite the radicand as a product of factors, looking specifically for factors that are perfect fourths.
\(\begin{align}
\sqrt[4]{{2\thinspace 592}} &= \sqrt[4]{{16\cdot 81 \cdot 2}} \\
&= \sqrt[4]{{2^4 \cdot 3^4 \cdot 2}} \\
\end{align}\)
Step 2: Rewrite the radical as a product of the perfect fourth radicals and the non-perfect fourth radicals.
\( \sqrt[4]{{2^4 \cdot 3^4 \cdot 2}} = \sqrt[4]{{2^4}}\cdot \sqrt[4]{{3^4}} \cdot \sqrt[4]{2}\)
Step 3: Simplify.
\( \sqrt[4]{{2^4}}\cdot \sqrt[4]{{3^4}} \cdot \sqrt[4]{2} = 2\cdot 3\sqrt[4]{2}\)
Therefore, \(\sqrt[4]{{2\thinspace 592}} = 6\sqrt[4]{2}\).
From this point on, because of its efficiency, the perfect \(n^{th}\) root factors method will be the only one used in solutions. However, you are certainly free to choose a solution method that works best for you.