Question
outline the steps involved in simplifying a square root to its simplest radical form.
Answers
GPT-4o mini
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
1. **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.
2. **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.
3. **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.
4. **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.
5. **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.
6. **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.
7. **Check for Further Simplification**:
- If the number inside the radical can be further simplified, repeat the steps as necessary.
### Example
Let's simplify \(\sqrt{72}\):
1. **Factor**: \( 72 = 2^3 \times 3^2 \)
2. **Group**: Pairs are \( (2^2) \) and \( (3^2) \)
3. **Take Square Roots**: From \( 2^2 \) take out \( 2 \) and from \( 3^2 \) take out \( 3 \)
4. **Multiply Outside**: \( 2 \times 3 = 6 \)
5. **Remaining Inside**: The remaining \( 2 \) from \( 2^3 \) is left inside.
6. **Combine**: \( 6\sqrt{2} \)
7. **Check**: \( 6\sqrt{2} \) can't be simplified further.
Thus, the simplest radical form of \(\sqrt{72}\) is \(6\sqrt{2}\).
### Steps for Simplifying a Square Root to Its Simplest Radical Form
1. **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.
2. **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.
3. **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.
4. **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.
5. **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.
6. **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.
7. **Check for Further Simplification**:
- If the number inside the radical can be further simplified, repeat the steps as necessary.
### Example
Let's simplify \(\sqrt{72}\):
1. **Factor**: \( 72 = 2^3 \times 3^2 \)
2. **Group**: Pairs are \( (2^2) \) and \( (3^2) \)
3. **Take Square Roots**: From \( 2^2 \) take out \( 2 \) and from \( 3^2 \) take out \( 3 \)
4. **Multiply Outside**: \( 2 \times 3 = 6 \)
5. **Remaining Inside**: The remaining \( 2 \) from \( 2^3 \) is left inside.
6. **Combine**: \( 6\sqrt{2} \)
7. **Check**: \( 6\sqrt{2} \) can't be simplified further.
Thus, the simplest radical form of \(\sqrt{72}\) is \(6\sqrt{2}\).