How do you simplify the square root of 180 and how do you know that it is in the simplest form

How do you simplify the square root of 180 and how do you know that it is in the simplest form

The Correct Answer and Explanation is:

To simplify the square root of 180, follow these steps:

1. Prime Factorization of 180:
   First, break down 180 into its prime factors. Start by dividing 180 by the smallest prime number, which is 2: 180÷2=90⇒180=2×90180 \div 2 = 90 \quad \Rightarrow \quad 180 = 2 \times 90180÷2=90⇒180=2×90 Next, divide 90 by 2 again: 90÷2=45⇒180=2×2×4590 \div 2 = 45 \quad \Rightarrow \quad 180 = 2 \times 2 \times 4590÷2=45⇒180=2×2×45 Now, divide 45 by 3, the next smallest prime number: 45÷3=15⇒180=2×2×3×1545 \div 3 = 15 \quad \Rightarrow \quad 180 = 2 \times 2 \times 3 \times 1545÷3=15⇒180=2×2×3×15 Divide 15 by 3: 15÷3=5⇒180=2×2×3×3×515 \div 3 = 5 \quad \Rightarrow \quad 180 = 2 \times 2 \times 3 \times 3 \times 515÷3=5⇒180=2×2×3×3×5 So, the prime factorization of 180 is: 180=22×32×5180 = 2^2 \times 3^2 \times 5180=22×32×5
2. Simplifying the Square Root:
   When simplifying a square root, look for pairs of identical factors. If there is a pair, take one factor out of the square root. From the prime factorization 180=22×32×5180 = 2^2 \times 3^2 \times 5180=22×32×5, we have:
   • A pair of 2’s (222^222)
   • A pair of 3’s (323^232)
   Each pair can be taken out of the square root as a single number. This means we can simplify the square root as follows: 180=22×32×5=22×32×5\sqrt{180} = \sqrt{2^2 \times 3^2 \times 5} = \sqrt{2^2} \times \sqrt{3^2} \times \sqrt{5}180​=22×32×5​=22​×32​×5​ Since the square root of a square is just the number itself: 22=2,32=3\sqrt{2^2} = 2, \quad \sqrt{3^2} = 322​=2,32​=3 So: 180=2×3×5=65\sqrt{180} = 2 \times 3 \times \sqrt{5} = 6\sqrt{5}180​=2×3×5​=65​
3. Verifying the Simplest Form:
   A square root is in its simplest form when no perfect square factors other than 1 are left inside the radical. In this case, 5 is not a perfect square, and there are no further simplifications. Therefore, the square root of 180 in its simplest form is 656\sqrt{5}65​.

Thus, the square root of 180 simplifies to 656\sqrt{5}65​, and it is in its simplest form because no further simplifications are possible.