Question
Simplyinf radicals
Square root of 567
Square root of 567
Answers
Answered by
GPT-4o mini
To simplify the square root of 567, we can start by factoring 567 into its prime factors.
1. **Factorization of 567:**
- 567 is odd, so we check for divisibility by 3:
- \( 567 \div 3 = 189 \)
- \( 189 \div 3 = 63 \)
- \( 63 \div 3 = 21 \)
- \( 21 \div 3 = 7 \)
- So, 567 can be expressed as:
\[
567 = 3^4 \times 7^1
\]
2. **Taking the square root:**
- The square root of a product of factors can be expressed as the product of the square roots of the factors:
\[
\sqrt{567} = \sqrt{3^4 \times 7} = \sqrt{3^4} \times \sqrt{7} = 3^2 \times \sqrt{7} = 9\sqrt{7}
\]
Thus, the simplified form of \( \sqrt{567} \) is \( 9\sqrt{7} \).
1. **Factorization of 567:**
- 567 is odd, so we check for divisibility by 3:
- \( 567 \div 3 = 189 \)
- \( 189 \div 3 = 63 \)
- \( 63 \div 3 = 21 \)
- \( 21 \div 3 = 7 \)
- So, 567 can be expressed as:
\[
567 = 3^4 \times 7^1
\]
2. **Taking the square root:**
- The square root of a product of factors can be expressed as the product of the square roots of the factors:
\[
\sqrt{567} = \sqrt{3^4 \times 7} = \sqrt{3^4} \times \sqrt{7} = 3^2 \times \sqrt{7} = 9\sqrt{7}
\]
Thus, the simplified form of \( \sqrt{567} \) is \( 9\sqrt{7} \).