Simplifying a square root to its simplest radical form involves a few systematic steps. Here’s an outline of the process:
Steps for Simplifying a Square Root to Its Simplest Radical Form
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Factor the Number Inside the Square Root:
- Start by finding the prime factorization of the number under the square root. This involves breaking the number down into its prime factors.
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Group the Factors:
- Identify pairs of the same factors. Since we are dealing with square roots, every pair will come out of the square root as a single number.
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Take Square Roots of the Pairs:
- For each pair of factors, take one factor out of the square root. If a factor appears three times, one can be left inside the square root while the other two come out.
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Multiply the Factors Outside the Square Root:
- Multiply all the factors that have been extracted from the square root together. This is your coefficient outside the radical.
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Multiply the Remaining Factors Inside the Square Root:
- If there are any factors left inside the square root after taking pairs out, multiply them together to form the radical part.
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Combine the Results:
- Write the simplified radical in the form of \( a\sqrt{b} \), where \( a \) is the coefficient outside the square root and \( b \) is the remaining value inside the square root.
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Check for Further Simplification:
- If the number inside the radical can be further simplified, repeat the steps as necessary.
Example
Let's simplify \(\sqrt{72}\):
- Factor: \( 72 = 2^3 \times 3^2 \)
- Group: Pairs are \( (2^2) \) and \( (3^2) \)
- Take Square Roots: From \( 2^2 \) take out \( 2 \) and from \( 3^2 \) take out \( 3 \)
- Multiply Outside: \( 2 \times 3 = 6 \)
- Remaining Inside: The remaining \( 2 \) from \( 2^3 \) is left inside.
- Combine: \( 6\sqrt{2} \)
- Check: \( 6\sqrt{2} \) can't be simplified further.
Thus, the simplest radical form of \(\sqrt{72}\) is \(6\sqrt{2}\).